Cascade of parametric instabilities in a multimode laser

PHYSICAL REVIEW A, VOLUME 64, 053808

Cascade of parametric instabilities in a multimode laser T. W. Carr* and T. Erneux Universite´ Libre de Bruxelles, Optique Nonline´aire The´orique, Campus Plaine, C. P. 231, 1050 Bruxelles, Belgium

C. Szwaj, M. Lefranc, D. Derozier, and S. Bielawski Laboratoire de Physique des Lasers, Atomes et Mole´cules, Universite´ des Sciences et Techniques de Lille, F-59655 Villeneuve d’Ascq Cedex, France 共Received 3 April 2001; published 4 October 2001兲 We analyze analytically and experimentally a cascade of parametric instabilities in an inhomogeneously broadened multimode laser under pump modulation. From the physical model we use a multiple-scale analysis to derive slow time and slow space amplitude equations, taking nonlocal coupling into account. Each of the two waves appearing at the primary instability acts as a pump for the parametric excitation of two new traveling waves with higher wave numbers. These predictions compare well with numerical simulations of the original laser equations and experimental results obtained with a modulated Yb-doped fiber laser. DOI: 10.1103/PhysRevA.64.053808

PACS number共s兲: 42.65.Sf, 47.54.⫹r

INTRODUCTION

Multimode laser instabilities have been the subject of many numerical investigations during the past decade 关1,2兴. However, combined analytical and experimental studies of their possible origins remain rare. In this paper, we concentrate on one of the simplest mechanisms, namely, a cascade of parametric instabilities that occurs in inhomogeneously broadened class-B lasers. An inhomogeneously broadened class-B laser, such as a fiber laser, can behave as a chain of coupled oscillators, each associated with one longitudinal mode 关3兴. It is therefore dynamically equivalent to a one-dimensional spatio-temporal system for which the mode index j 共or equivalently the optical wavelength兲 plays the role of the spatial variable 关3,4兴. Damped waves 共called ‘‘spectral waves’’兲 may propagate in this set of modes 共and thus in the optical spectrum兲 关3兴 and can be parametrically excited by modulating the pump 关4兴. They admit the typical bifurcation properties of parametrically excited systems, such as the Faraday experiment in hydrodynamics 关5兴 and plasmas 关6兴. The laser system exhibits two unusual features that make it particularly interesting for analytical, numerical, and experimental studies. First, each laser mode is coupled to a large number of other modes 共called finite range coupling or nonlocal coupling兲 关1,7,3兴. Second, the laser admits a decreasing dispersion curve for the spectral traveling waves. There is currently a large interest for dynamical systems showing finite range coupling 关8,9兴 as well as devices with unusual dispersion properties 关10兴. General conclusions or their effects are, however, harder to determine. For instance, we may be tempted to artificially add a term modeling finite range coupling to an already existing amplitude equation 共such as a Ginzburg-Landau equation 关8兴兲. But extracting amplitude equations from the original equations exhibiting finite range coupling remains a difficult project when the

*Permanent address: Department of Mathematics, Southern Methodist University, Dallas, Texas 75275. 1050-2947/2001/64共5兲/053808共9兲/$20.00

coupling is not diffusive. Indeed, if the coupling between modes depends on the original space scale, we intuitively expect that its effect on the long space scale will be local. As we shall demonstrate, this idea is correct for all periodic traveling wave modes but not for the uniform time-periodic mode. In this paper we propose an analysis of laser rate equations modeling an inhomogeneously broadened laser subject to a modulated pump. We show that a cascade of bifurcations results from the particular dispersion curve as well as from the finite range coupling. Each bifurcation corresponds to a particular combination of spectral modes and we determine slow time and long space amplitude equations for their respective evolution. In contrast to our earlier work on the problem 关4兴, the objectives of this paper are twofold. First, we resolve a multiple scale difficulty which was ignored in 关4兴 and properly derive the amplitude equations. Second, we consider a higher number of resonant modes leading to higher order bifurcations. This bifurcation analysis is motivated by our recent experiments which clearly show a gradual increase of the number of spectral modes as we progressively increase the modulation amplitude from zero. The plan of the paper is as follows. In Sec. I we formulate the laser rate equations and take advantage of the laser natural range of parameters. In Sec. II we construct small amplitude traveling wave solutions and derive their amplitude equations. In Sec. III we analyze the first two bifurcations. They provide a guide for our numerical simulations of the laser original equations, summarized in Sec. IV. Finally, we discuss in Sec. V experimental results obtained by using a Yb-doped fiber laser. I. FORMULATION

We consider an inhomogeneously broadened class-B laser without phase-sensitive interactions. The state of such a laser can be described by a set of mode intensities s j (t) and a continuous set of population inversion d(x,t) 关4,1,7兴. In dimensionless form and considering the continuous limit of a large number of modes 关i.e., s j (t) is replaced by s(x,t)兴,

64 053808-1

©2001 The American Physical Society

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PHYSICAL REVIEW A 64 053808

these equations take a simple form given by 关4兴

⳵ t s 共 x,t 兲 ⫽⫺s 共 x,t 兲 ⫹ 关 s 共 x,t 兲 ⫹a 兴

冋

冕

⬁

⫺⬁

␤ 共 y⫺x 兲 d 共 y,t 兲 dy,

冊册

共1a兲

冉 冕

⳵ t d 共 x,t 兲 ⫽ ␥ g 共 x 兲 A 共 t 兲 ⫺d 共 x,t 兲 1⫹ ⫻s 共 y,t 兲 dy

␧⬅

⬁

⫺⬁

␤ 共 y⫺x 兲 共1b兲

.

In these equations, ␤ (x) is a cross-saturation coupling coefficient defined by

␤共 x 兲⬅

1 1 ␲ 1⫹x 2

共2兲

and its Lorentzian shape ensures the coupling between the modes. The spatial coordinate x refers to an emission optical frequency of a specific mode. Note that the intensities and populations evolve according to much slower time scale than the optical ones. Time t is measured in units of the cavity lifetime ␶ c which is typically in the microsecond range. The parameter a is the spontaneous-emission coefficient; it will be neglected in the analytical calculation, but not in the numerical simulation. The parameter ␥ is the population inverand is sion rate normalized by the photon decay rate ␶ ⫺1 c typically small. A(t)⫽A 关 1⫹m cos(␻mt)兴 denotes the modulated pumping rate where A, m, and ␻ m represent its average, amplitude, and frequency, respectively. ␻ m and m are our two control parameters and ␻ m is chosen close to the relaxation oscillation frequency of the single mode laser defined by ␻ r ⬅ 冑␥ (A⫺1). Nonuniformities of the laser spectrum are modeled by the selective pumping coefficient g(x). In our analytical work, we shall consider an infinitely large medium with g(x)⫽1. In our numerical simulations, we use a Gaussian function for g(x). We may rewrite Eqs. 共1a兲 and 共1b兲 in a more elegant form for analysis by introducing deviations from the uniform steady state, (s,d)⫽(A⫺1,1), and by rescaling time with the relaxation oscillations frequency ␻ r . Specifically, we introduce the variables u, v , and ␶ defined by u⫽⫺1⫹

s , A⫺1

v⫽

d⫺1 , ␻r

and ␶ ⫽ ␻ r t

共3兲

into Eqs. 共1a兲 and 共1b兲 and obtain

⳵ ␶ u⫽ 共 1⫹u 兲

冕

⬁

⫺⬁

⳵ ␶ v ⫽⫺ 共 1⫹2␧F ⫺1 v 兲

冕

␤ 共 y⫺x 兲v共 y 兲 dy,

⬁

⫺⬁

␦⬅

Am , A⫺1

F⬅

A , A⫺1

and ␴ ⬅

␻m . 共5兲 ␻r

In Eq. 共4b兲, all parameters are O(1) except the dissipation parameter ␧ and the forcing amplitude ␦ . The small parameter ␧ measures the natural damping of the laser relaxation oscillations in units of the relaxation frequency ␻ r . Since ␻ r is O( 冑␥ ) small, ␧ is an O( 冑␥ ) small quantity. ␦ is proportional to the modulation amplitude m, and ␴ is the modulation frequency normalized by the relaxation oscillation frequency ␻ r . It is worthwhile to briefly describe the results of the linear stability analysis in the case of no pump modulations ( ␦ ⫽0) 关3兴. Introducing u⫽c 1 exp(ikx)exp(␭␶) and v ⫽c 2 exp(ikx)exp(␭␶) into the linearized equations leads to the characteristic equation ␭ 2 ⫹2␧␭⫹B 2 ⫽0, where B(k) ⫽exp(⫺兩k兩) is the Fourier transform of ␤ (x). For small ␧, ␭ admits the following approximation: ␭⫽⫾i ␻ ⫺␧⫹O 共 ␧ 2 兲 ,

共6兲

where ␻ is given by

␻ ⫽B 共 k 兲 ⫽exp共 ⫺ 兩 k 兩 兲 .

共7兲

Equation 共7兲 is the leading expression of the dispersion relation ␻ ⫽ ␻ (k) for the periodic traveling waves of the form u⫽c 1 exp(i␻␶⫾ikx) and v ⫽c 2 exp(i␻␶⫾ikx). It will reappear in our bifurcation analysis. We wish to determine the solution generated by near resonance modulations ( ␴ ⬃1). A fundamental parametric instability similar to Faraday instability in hydrodynamics may then occur if ␻ ⬃1/2. This instability leads to a period doubling bifurcation to a stationary wave which we have studied numerically and experimentally in 关4兴. The wave number k 1 of this stationary wave verifies the dispersion relation 共7兲. But other traveling wave solutions exhibiting other frequencies and wave numbers have been observed experimentally. Our goal is to show that these waves do not appear by chance but sequentially appear through higher order bifurcations. To this end, we develop a multiple scale analysis of the laser equations 共4a兲 and 共4b兲 and reduce the original laser equations to amplitude equations which are easier to investigate. II. MULTIPLE-SCALE ANALYSIS

We wish to determine the solution, Eqs. 共4a兲 and 共4b兲, for ␴ close to 1 and ␦ small. To this end, we first expand these parameters as

␴ ⫽1⫹␧ ␴ 1 ⫹•••,

共4a兲

␦ ⫽␧ 2 ␦ 2 ⫹•••,

共8兲

where ␧ is defined in Eq. 共5兲. The scaling for ␦ is imposed by the solvability conditions. We then seek a solution of the form

␤ 共 y⫺x 兲 u 共 y 兲 dy⫺2␧ v

⫹ ␦ cos共 ␴ ␶ 兲 ,

␥A , 2␻r

共4b兲

where new parameters ␧, ␦ , ␴ are 053808-2

u⫽␧u 1 共 T, ␪ ,x, ␨ 兲 ⫹␧ 2 u 2 共 T, ␪ ,x, ␨ 兲 ⫹•••,

共9a兲

v ⫽␧ v 1 共 T, ␪ ,x, ␨ 兲 ⫹␧ 2 v 2 共 T, ␪ ,x, ␨ 兲 ⫹•••,

共9b兲

CASCADE OF PARAMETRIC INSTABILITIES IN A . . .

PHYSICAL REVIEW A 64 053808 TABLE I. Parametric resonances. The primary resonance involves the first three modes and verifies the conditions 共12兲. The second resonance is realized with the four remaining modes and verifies the conditions 共12兲–共14兲.

FIG. 1. Diagram showing the cascade process.

where T⬅ ␴ ␶ is our new basic time, ␪ ⬅␧ ␶ is a slow time variable, and ␨ ⬅␧x is a long space variable. The two independent time scales require the chain rule u ␶ ⫽ ␴ u T ⫹␧u ␪ . The assumption of two distinct space variables needs, however, a careful treatment because the spatial coupling between the modes is realized through an integral operator rather than a differential operator. All mathematical details are relegated to the Appendixes and we summarize the main results. The leading order problem for u 1 and v 1 is linear and admits a multimode solution of the form

兺n a n共 ␪ , ␨ 兲 exp共 i⍀ n T⫹ik n x 兲 ⫹c.c.,

共10兲

兺n ia n共 ␪ , ␨ 兲 exp共 i⍀ n T⫹ik n x 兲 ⫹c.c.,

共11兲

u 1⫽ v 1⫽

provided that all frequencies ␻ ⫽⍀ n and wave numbers k ⫽k n satisfy the dispersion relation 共7兲. c.c. means complex conjugate. We shall consider the interaction of seven modes which appear through a cascade of parametric instabilities illustrated in Fig. 1. Specifically, the uniform mode of amplitude a 0 excites a stationary wave made of two counterpropagating traveling waves of amplitudes a 1 and a 2 , respectively. Next, the traveling wave of amplitude a l excites the pair of traveling waves of amplitudes a ll and a lm 共with l and m⫽1,2). For example, the amplitude a 1 excites the pair of traveling waves of amplitude a 11 and a 12 . This notation allows us to identify the origin of a n 共n⫽0,1,2,11,12,21,22兲, and will be useful as we examine the symmetries in the amplitude equations. The first parametric instability is realized by the resonance condition k 1 ⫹k 2 ⫽0

and ⍀ 1 ⫽⍀ 2 ⫽1/2.

k 11⫹k 12⫽k 1 , k 21⫹k 22⫽⫺k 1

⍀ 11⫹⍀ 12⫽⍀ 1 ⫽1/2,

共13兲

and ⍀ 21⫹⍀ 22⫽⍀ 2 ⫽1/2.

共14兲

Because the amplitudes a n in Eqs. 共10兲 and 共11兲 are unknown, we need to investigate the next problem for u 2 and v 2 and apply solvability conditions. These conditions lead to equations for the amplitudes a n which are given by Eqs. 共A13兲–共A19兲 in Appendix A. They exhibit both sumfrequency terms S n ⫽a i a j as well as parametric gain terms P n ⫽a i a *j thanks to the resonance conditions 共12兲–共14兲. Be-

⍀n

kn

0 1 2 11 12 21 22

1 1/2 1/2 1/6 1/3 1/3 1/6

0 ⫺ln(2) ln(2) ⫺ln(6) ln(3) ⫺ln(3) ln(6)

fore we analyze these equations, we evaluate all wave numbers and frequencies using the conditions 共7兲 and 共12兲–共14兲. They are listed in Table I for clarity. We have verified that the linear terms appearing in the amplitude equations 共A13兲– 共A19兲 reproduce the informations from the linear stability analysis, i.e., the growth rate in powers of ␧ given by Eq. 共6兲 共see Appendix C兲. These linear terms provide the O(␧) changes of each resonant wave number and each resonant frequency resulting from the detuning ␴ 1 . It will be physically and mathematically useful to take into account these changes and reformulate the leading approximation of the solution. Specifically, we introduce the new amplitude b n ⫽a n exp共 ⫺i ␮ n ␪ ⫺i ␬ n ␨ 兲 共 n⫽0 兲

共15兲

into Eqs. 共10兲 and 共11兲 and require that ␮ n and ␬ n satisfy the O(␧) contribution of the dispersion relation 共7兲. Noting that the O(␧) correction of the frequency measured in units of time ␶ is ␮ n ⫹⍀ n ␴ 1 , this relation is given by

␮ n ⫹⍀ n ␴ 1 ⫽ v gn ␬ n

for n⫽0,

共16兲

where v gn is the group velocity defined by Eq. 共A20兲. We may then rewrite the solutions 共10兲 and 共11兲 as

兺n b n共 ␪ , ␨ 兲 exp共 i ␻ˆ n ␶ ⫹ikˆ n x 兲 ⫹c.c.,

共17兲

兺n ib n共 ␪ , ␨ 兲 exp共 i ␻ˆ n ␶ ⫹ikˆ n x 兲 ⫹c.c.,

共18兲

u 1⫽

v 1⫽

共12兲

The second parametric instability is possible if

n

ˆ n (␧)⬅⍀ n ⫹␧( ␮ n ⫹⍀ n ␴ 1 ) and kˆ n (␧)⬅k n ⫹␧ ␬ n where ␻ represent the effective frequencies and wave number associated with each traveling wave mode and b 0 ⫽a 0 , ␮ 0 ⫽ ␬ 0 ⫽0. The main benefit of Eqs. 共17兲 and 共18兲 is the fact that a constant b n now corresponds to a traveling wave mode amplitude at the effective frequencies and wave number. Moreover, the evolution equations for the b n are simpler than the equations for the a n . Of course, we also wish that the effective frequencies and wave numbers satisfy the parametric ˆ resonance conditions. From Eqs. 共12兲–共14兲 with ⍀ n ⫺1 ˆ ⫽ ␴ ␻ n replacing ⍀ n and kˆ n replacing k n , respectively, we obtain the following conditions for ␮ n and ␬ n :

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T. W. CARR et al.

PHYSICAL REVIEW A 64 053808

␬ 1 ⫹ ␬ 2 ⫽0, ␬ 11⫹ ␬ 12⫽ ␬ 1 , ␬ 21⫹ ␬ 22⫽⫺ ␬ 1 ,

␮ 1 ⫽ ␮ 2 ⫽0, ␮ 11⫹ ␮ 12⫽0, ␮ 21⫹ ␮ 22⫽0.

共19兲 共20兲 共21兲

Together with Eq. 共16兲, we easily determine ␬ n and ␮ n as

␮ 1 ⫽ ␮ 2 ⫽0,

共22兲

␬ 1 ⫽⫺ ␬ 2 ⫽ ␴ 1 ,

共23兲

␴1 ␮ 11⫽ ␮ 22⫽⫺ ␮ 12⫽⫺ ␮ 21⫽ , 9 5␴1 , ␬ 11⫽⫺ ␬ 22⫽ 3

2␴1 . ␬ 12⫽⫺ ␬ 21⫽⫺ 3

共24兲

共33兲

.

It represents a uniform 2 ␲ / ␻ m time-periodic solution of the original laser equations 共1a兲 and 共1b兲. We next determine a primary bifurcation from this spatially uniform solution to a stationary wave solution. The stationary wave solution satisfies the condition b n ⫽0 (n⫽11,12,21,22). We first solve the equations for b 1 and b 2 and then the equation for b 0 and obtain 兩 b 1兩 ⫽ 兩 b 2兩 .

共25兲

共26兲

i b˙ 1 ⫺ v g1 b ⬘1 ⫽⫺b 1 ⫹ 共 b 0 b 2* ⫹b 11b 12兲 , 4

共27兲

i b˙ 2 ⫺ v g2 b 2⬘ ⫽⫺b 2 ⫹ 共 b 0 b 1* ⫹b 21b 22兲 , 4

共28兲

i b b* , 12 1 12

共29兲

i *, ⬘ ⫽⫺b 12⫹ b 1 b 11 b˙ 12⫺ v g12b 12 6

共30兲

i *, ⬘ ⫽⫺b 21⫹ b 2 b 22 b˙ 21⫺ v g21b 21 6

共31兲

i b b* , 12 2 21

共32兲

共34兲

兩 b 1 ( ␦ 22 ) 兩 is given by the implicit solution

␦ 22 ⫺ ␦ 2P ⫽64兩 b 1 兩 2 共 1⫹ 161 兩 b 1 兩 2 兲 ,

i␦2 i ⫹ b 1b 2 , b˙ 0 ⫺iH 共 b 0⬘ 兲 ⫽⫺b 0 ⫺i ␴ 1 b 0 ⫺ 4 2

⬘ ⫽⫺b 22⫹ b˙ 22⫺ v g22b 22

␦2

4 冑1⫹ ␴ 21

兩 b 0 兩 ⫽4,

Using Eq. 共15兲 with Eqs. 共22兲–共25兲, our amplitude equations 共A13兲–共A19兲 simplify as

⬘ ⫽⫺b 11⫹ b˙ 11⫺ v g11b 11

兩 b 0兩 ⫽

共35兲

where ␦ P is a primary bifurcation point defined by

␦ P ⬅4 冑1⫹ ␴ 21 .

共36兲

The bifurcation is clearly supercritical 共i.e., ␦ 2 ⬎ ␦ P if 兩 b 1 兩 ⫽0) and leads to a 4 ␲ / ␻ m time-periodic stationary wave solution of the original laser equations which we analyzed in 关4兴. But this solution may itself exhibit a secondary bifurcation to a more complex solution with now all the b n ⫽0. We determine this new solution by solving first the equations for b 11 , b 12 , b 21 , b 22 , then the equations for b 1 and b 2 , and finally the equation for b 0 . We find 兩 b 1 兩 ⫽ 兩 b 2 兩 ⫽6 冑2,

共37兲

兩 b 12兩 ⫽ 兩 b 21兩 ⫽2 兩 b 11兩 ⫽2 兩 b 22兩 ,

共38兲

where 兩 b 11兩 is related to 兩 b 0 兩 by 兩 b 11兩 ⫽2 冑6

冉

冊

兩 b 0兩 ⫺1 . 4

共39兲

Finally 兩 b 0 ( ␦ 22 ) 兩 ⭓4 is given by the implicit solution

␦ 22 ⫺ ␦ 2S ⫽16共 兩 b 0 兩 ⫺4 兲关共 兩 b 0 兩 ⫹4 兲共 1⫹ ␴ 21 兲 ⫹72兴 ,

共40兲

where ␦ S is a secondary bifurcation point defined by

␦ S ⫽16冑100⫹ ␴ 21 .

共41兲

where the ␴ 1 term do not appear in the equation for b n (n ⫽0). In these equations, dot and prime mean partial derivatives with respect to time ␪ and space ␨ , respectively. v gn is the group velocity defined by Eq. 共A20兲 and H(b 0⬘ ) represent the Hilbert transform of b 0⬘ defined by Eq. 共A6兲.

The bifurcation is clearly supercritical since 兩 b 0 兩 ⭓4. It leads to a multifrequency traveling wave solution of the original ˆ n (␧) shows laser equations. Computing the frequencies ␻ that this solution is not periodic if ␴ 1 ⫽0.

III. CASCADING BIFURCATIONS

IV. NUMERICAL STUDY

In this section we determine the bifurcation diagram of the steady and uniform solutions of Eqs. 共26兲–共32兲. The analysis is long and tedious but leads to simple analytical results which we summarize. The basic solution verifies the condition b n ⫽0 (n⫽0) and admits the amplitude

Our previous analysis of the laser equations 共1a兲 and 共1b兲 with g(x)⫽1 and a⫽0 led to simple analytical expressions for the primary and secondary bifurcations. They will be useful as we integrate Eqs. 共1b兲 and 共1b兲 numerically with a spatial variation of the pump and nonzero values of the spontaneous emission coefficient. Specifically, we consider g(x)

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CASCADE OF PARAMETRIC INSTABILITIES IN A . . .

PHYSICAL REVIEW A 64 053808

FIG. 2. Numerical solution of the laser equations 共1a兲 and 共1b兲 after the primary bifurcation ( ␻ m ⫽0.80␻ r and m⫽0.16).

⫽exp(⫺x2/2␳ 2 ) where the width of the Gaussian ␳ is changed from 70 to 10, and the values of a from 10⫺3 to 10⫺5 . We also investigated the change of the bifurcation diagram as the laser fixed parameter A and ␥ are changed 关 A 共several unities兲 and ␥ (10⫺3 to 10⫺5 )兴. The values of the laser fixed parameters that best correspond to our experiments are given by A⫽3, ␥ ⫽1.34 ⫻10⫺4 , a⫽8.3⫻10⫺4 , ␳ ⫽75/7, and ␻ m ⫽0.8␻ r . The

FIG. 3. 共a兲 Numerical solution of the laser equations 共1a兲 and 共1b兲 after the secondary bifurcation ( ␻ m ⫽0.8␻ r and m⫽0.195); 共b兲 high temporal frequency part showing the basic stationary wave; 共c兲 low temporal frequency part showing the new waves emerging at the secondary bifurcation.

FIG. 4. Temporal and spatial Fourier transform of the solution shown in Fig. 3共a兲. The dispersion curve, given by Eq. 共7兲, is shown in the figure by a dashed line.

modulation amplitude m is the bifurcation parameter. As we progressively increase m from zero, we typically observe the Faraday instability as the primary instability. This bifurcation leads to a subharmonic stationary wave 共Fig. 2兲, as previously reported in Ref. 关4兴. As we further increase m, this solution changes stability through a secondary bifurcation. The new solution exhibits additional modulations in space and time 共Fig. 3兲. In order to clearly identify the new components of the secondary bifurcating solution and verify that they appear through the mechanism predicted in Sec. III, it is worth examining the Fourier transforms both in space and time of the spectrochronogram of Fig. 3共a兲 共Fig. 4兲. As can be seen in Fig. 4, we observe six main spectral components in addition to the fundamental component 共located at ␻ m ⫽ ␻ r ,k⫽0兲. The two first are localized at the subharmonics of the modulation and correspond to the usual Faraday waves ( ␻ 1 ⫽ ␻ 2 ⫽ ␻ m /2,k 1 ⫽⫺k 2 ). The four other components appear at the secondary bifurcation and correspond to four traveling waves. Furthermore, one pair of traveling waves 兵 ( ␻ 21 ,k 21),( ␻ 22 ,k 22) 其 satisfies the resonance conditions k 2 ⫽k 21⫹k 22 ,

共42a兲

␻ 2 ⫽ ␻ 21⫹ ␻ 22 ,

共42b兲

and a similar relation is verified for the other pair of traveling waves 兵 ( ␻ 11 ,k 11),( ␻ 12 ,k 12) 其 . Graphically 共see Fig. 4兲, this relation means that we may draw a parallelogram in the ( ␻ ,k) diagram which connects the origin, one point associated to the Faraday mode, and two points associated to the new modes appearing at the secondary instability. This parallelogram clearly indicates that we are in presence of the secondary bifurcating solution predicted by our analysis. High-frequency temporal filtering ( ␻ ⬎0.4␻ m ) allows us to extract and visualize the high-frequency part 关i.e., the basic Faraday stationary wave, Fig. 3共b兲兴. Low-frequency filtering ( ␻ ⬍0.4␻ m ) allows us to keep only the four traveling

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PHYSICAL REVIEW A 64 053808

FIG. 5. Experimental setup. The laser cavity is delimited by the R max mirror (M 2 ), and the air/glass interface (M 1 ) at the extremity of the undoped fiber 共UF兲. Yb is the Yb3⫹ -doped fiber; (L1) and (L2) are, respectively, a collimating lens, and a microscope objective (⫻10); the dichroic mirror 共DM兲 is R max at the laser wavelength.

waves which exhibit the lowest frequencies and which appear at the secondary bifurcation. Their superposition leads to the pattern shown in 关Fig. 3共c兲兴.

FIG. 6. Experimental regime observed after the primary bifurcation (m⫽0.18).

allows us to visualize the Faraday stationary wave while the low temporal filtering reveals the secondary bifurcating traveling wave modes 关Fig. 7共c兲兴. At much higher modulation amplitudes, the quasiperiodic regime becomes unstable and we observe more complex 共chaotic兲 regimes which we do not analyze.

V. EXPERIMENTAL STUDY A. Experimental setup

VI. SUMMARY

The system under investigation is a Yb-doped fiber laser operating at 1.04 ␮ m. The fiber is cooled at 77 K in order to obtain a small homogeneous width. Thus we obtain a large aspect ratio, as is supposed in the theoretical study 关11兴. The experimental setup is represented in Fig. 5. The active Yb3⫹ silica fiber 共CNET FPGA461Yb兲 is fusion spliced to 2 m of an undoped one. Both fibers are single mode at pump and laser wavelengths and the overall length is 19 m. The pump is provided by a single mode 852 nm diode laser 共SDL5411兲. The input mirror is constituted by the glass/air interface (R ⫽4%) at the input, and the output mirror is a Rmax around 1.04 ␮ m. Under typical conditions, the output power can typically reach 15 mW. The optical spectrum is analyzed by means of a monochromator followed by a Silicium chargecoupled device 共CCD兲 array 共EGG RL0256D兲. One spectrum is recorded at each period of modulation by a Datel PCI416H acquisition board, and we perform a real-time visualization of the spectrochronograms during the experiment. In the following, the pump rate is A⫽3, and the modulation frequency ␻ m ⫽0.8␻ r , with ␻ r the relaxation frequency of the laser ␻ r ⫽40 kHz.

In this paper we proposed a combined analytical, numerical, and experimental study of an inhomogeneously broad-

B. Results

Experimentally, when we increase the modulation amplitude from zero, we observe the same bifurcations scenario as noted numerically. First, a supercritical bifurcation occurs leading to a stationary wave inside the spectrum 共Fig. 6兲. This corresponds to the primary bifurcation 共or Faraday instability兲. When the modulation amplitude is further increased, we observe a secondary bifurcation leading to a multifrequency regime 关Fig. 7共a兲兴. We have verified that the Fourier transform in space and time of this regime displays the six spectral components that characterizes the secondary bifurcation 共Fig. 8兲. As in our numerical study, high frequency temporal filtering 关Fig. 7共b兲兴

FIG. 7. 共a兲 Experimental regime observed after the secondary bifurcation (m⫽0.20). 共b兲 High temporal frequency part showing the basic stationary wave. 共c兲 Low temporal frequency part showing the new waves appearing after the secondary bifurcation.

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ser two first bifurcations leading to two distinct patterns that compare well with the solutions obtained numerically. ACKNOWLEDGMENTS

We would like to thank Dominique Razafimahatratra and Philippe Verkerk for their contributions in the experiment. The Laboratoire de Physique des Lasers, Atomes et Molecules is Unite´ de Recherche Mixte du CNRS. The Center d’E´tudes et de Recherches Lasers et Applications is supported by the Ministe`re charge´ de la Recherche, the Re´gion Nord-Pas de Calais, and the Fonds Europe´en de De´veloppement E´conomique des Re´gions. T.W.C. was supported by the National Science Foundation Grant No. DMS-9803207. The research by T.E. was supported by the U.S. Air Force Office of Scientific Research Grant No. AFOSR F49620-98-1-0400, the National Science Foundation Grant No. DMS-9973203, the Fonds National de la Recherche Scientifique 共Belgium兲, and the InterUniversity Attraction Pole program of the Belgian government. FIG. 8. Temporal and spatial Fourier transform of the pattern shown in Fig. 7共a兲.

ened multimode fiber laser. We concentrated on the first two bifurcations because they are immediately followed by more complex 共chaotic兲 regimes as the modulation amplitude is further increased. The analysis of the laser equations considers the simple case of a uniform pump 关 g(x)⫽1 兴 . We derived amplitude equations for the resonant traveling wave modes that appear at the primary and secondary bifurcation points. Of particular interest is the nonlocal coupling term 共the Hilbert transform兲 that appears for the uniform periodic mode. It models the effect of finite range coupling now on the long space scale ␨ ⫽␧x. It is interesting to note that the same Hilbert transform appears in a single partial differential equation describing a planar flame front subject to Darrieus-Landau instability 关12,13兴. The main advantage of our amplitude equations is the fact that steady states solutions of these equations correspond to traveling wave solutions of the original laser equations. Consequently, we obtained useful expressions of the bifurcation points in term of the laser parameters. Other spatio-temporal solutions of these equations may be possible but were not suggested by our numerical study of the laser equations. If we consider g(x) as a slowly varying function of ␨ 关i.e., we consider ␳ in g(x)⫽exp(⫺x2/2␳ 2 ) as an O(␧ ⫺1 ) large parameter and reformulate g(x) as g⫽g( ␨ )], amplitude equations similar to Eqs. 共26兲–共32兲 can still be derived but exhibit coefficients which are functions of ␨ . Consequently, constant solutions for the a n are no longer possible and their equations need to be studied numerically. This problem is currently investigated. The numerical study of the original laser equations considers the realistic case of g(x) given by a Gaussian function of x. We have found that the first two bifurcations correspond to the bifurcations documented analytically for g(x)⫽1. In particular, we identified the main features of the primary and secondary bifurcating solutions by examining their spectral properties in detail. Finally, we found experimentally the la-

APPENDIX A

The integral operator F 共 u,␧ 兲 ⫽

1 ␲

冕

⬁

1

⫺⬁ 1⫹ 共 y⫺x 兲 2

u 共 y,␧y 兲 dy

共A1兲

depends on ␧ because we assume that u is a function of two independent spatial variables x and ␨ ⫽␧x. A naive two terms expansion of the integral 共A1兲 gives F 共 u,␧ 兲 ⫽

1 ␲

冕

⬁

1

⫺⬁ 1⫹ 共 y⫺x 兲 2

u„y,␧x⫹␧ 共 y⫺x 兲 …dy 共A2兲

⫽F 0 共 u 兲 ⫹␧F 1 共 u 兲 ⫹•••,

共A3兲

where F 0共 u 兲 ⫽

1 ␲

F 1共 u 兲 ⫽

1 ␲

冕

⬁

1

⫺⬁ 1⫹ 共 y⫺x 兲 2

冕

⬁

共 y⫺x 兲

⫺⬁ 1⫹ 共 y⫺x 兲 2

u 共 y, ␨ 兲 dy,

共A4兲

u ␨ 共 y, ␨ 兲 dy.

共A5兲

Equation 共A4兲 is the original operator evaluated at ␧y⫽␧x while Eq. 共A5兲 is a new integral acting on u ␨ . If u(y, ␨ ) is only a function of ␨ and if u ␨ is assumed constant as we integrate with respect to the fast spatial variable y, the integral 共A5兲 is clearly unbounded. A more careful treatment is needed for this case and leads to a Hilbert transform given by 共see Appendix B兲 F 1 共 u 兲 ⫽H 共 u ␨ 兲 ⫽

1 ␲

冕

⬁

1 u ␰共 ␰ 兲 d ␰ . ⫺⬁ ␰ ⫺ ␨

共A6兲

In our perturbation analysis, we apply the integral operator 共A1兲 when u is a sum of periodic functions of the form

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PHYSICAL REVIEW A 64 053808

共A7兲

i i *, ⬘ ⫽⫺a 11⫺ ␴ 1 a 11⫹ a 1 a 12 a˙ 11⫺ v g11a 11 6 12

共A16兲

Using Eq. 共A7兲, we may evaluate F 0 and F 1 and obtain the useful formula

i i *, ⬘ ⫽⫺a 12⫺ ␴ 1 a 12⫹ a 1 a 11 a˙ 12⫺ v g12a 12 3 6

共A17兲

i i *, ⬘ ⫽⫺a 21⫺ ␴ 1 a 21⫹ a 2 a 22 a˙ 21⫺ v g21a 21 3 6

共A18兲

i i *. ⬘ ⫽⫺a 22⫺ ␴ 1 a 22⫹ a 2 a 21 a˙ 22⫺ v g22a 22 6 12

共A19兲

u⫽

F 0共 u 兲 ⫽

F 1共 u 兲 ⫽

兺

m⫽0

兺m a m共 ␨ 兲 exp共 ik m x 兲 .

兺m a m共 ␨ 兲 exp共 ⫺ 兩 k m兩 兲 exp共 ik m x 兲 ,

共A8兲

⬘ 共 ␨ 兲 exp共 ⫺ 兩 k m 兩 兲 exp共 ik m x 兲 sgn共 k m 兲 ia m

⫹H„a 0⬘ 共 ␨ 兲 …,

共A9兲

where H(a 0⬘ ) is the Hilbert transform of a 0⬘ , and the prime means derivation with respect to ␨ . We are now ready to start our perturbation analysis. Introducing Eqs. 共8兲–共9b兲 into Eqs. 共4a兲 and 共4b兲 leads to the following sequence of problems for (u 1 , v 1 ) and (u 2 , v 2 ): for O(␧) u 1T ⫽F 0 共 v 1 兲

and v 1T ⫽⫺F 0 共 u 1 兲 .

共A10兲

In these equations, the dots and primes mean partial derivatives with respect to time ␪ and space ␨ , respectively. The coefficient v gn is the group velocity defined as v gn ⬅

冋 册 dB 共 k 兲 dk

共A20兲

and H(a ⬘0 ) represents the Hilbert transform of a ⬘0 ( ␨ ). APPENDIX B

For O(␧ 2 ) u 2T ⫺F 0 共 v 2 兲 ⫽R 1 ⬅u 1 F 0 共 v 1 兲 ⫺ ␴ 1 u 1T ⫺u 1 ␪ ⫹F 1 共 v 1 兲 ,

We wish to find the limit of the integral 共A11兲

I共 x 兲⫽

v 2T ⫹F 0 共 u 2 兲 ⫽R 2

⬅⫺2 v 1 ⫹ ␦ 2 cos共 T 兲 ⫺ ␴ 1 v 1T ⫺ v 1 ␪ ⫺F 1 共 u 1 兲 . The solution of Eq. 共A10兲 is a sum of periodic plane wave solutions given by 共10兲 and 共11兲. Because the amplitude a n ( ␪ , ␨ ) in 共10兲 and 共11兲 are unknown, we consider the higher order problem given by Eq. 共A11兲 and apply solvability conditions with respect to each critical mode. Taking the T derivative of Eq. 共A11兲 and using the equation for u 2 , we obtain an equation for v 2 only. The solvability conditions are then the following orthogonality conditions: 0⫽

⫽⫺⍀ n sgn共 k n 兲 k⫽k n

冕 冕 2 ␲ /⍀ m

0

2 ␲ /k m

0

冕

⬁

1

⫺⬁ 1⫹ 共 y⫺x 兲 2

u 共 ␧y 兲 dy

共B1兲

as ␧→0. In order to find the correct limit, we shall consider the Fourier transform of Eq. 共B1兲 and expand its expression in power series of ␧. To this end, we introduce the coordinates X⬅␧x and Y ⬅␧y and rewrite Eq. 共B1兲 as I共 x 兲⫽

␧ ␲

冕

⬁

1

⫺⬁ ␧ 2 ⫹ 共 Y ⫺X 兲 2

u 共 Y 兲 dY

共B2兲

so that ␧ appears explicitly in the integral. The Fourier transform is defined by the integral 关14兴

exp共 ⫺i⍀ n T⫺ik n x 兲关 R 2T ⫺F 0 共 R 1 兲兴

⫻dTdx.

1 ␲

F共 ␰ 兲⫽

共A12兲

冕 冑 ␲ 1

⬁

2

⫺⬁

f 共 x 兲 exp共 i ␰ x 兲 dx

共B3兲

F 共 ␰ 兲 exp共 ⫺i ␰ x 兲 d ␰ .

共B4兲

and the inversion integral is These conditions lead to seven coupled partial differential equations for the amplitudes a n given by i␦2 i ⫹ a 1a 2 , a˙ 0 ⫺iH 共 a 0⬘ 兲 ⫽⫺a 0 ⫺i ␴ 1 a 0 ⫺ 4 2 i i a˙ 1 ⫺ v g1 a ⬘1 ⫽⫺a 1 ⫺ ␴ 1 a 1 ⫹ 共 a 0 a 2* ⫹a 11a 12兲 , 2 4 i i a˙ 2 ⫺ v g2 a ⬘2 ⫽⫺a 2 ⫺ ␴ 1 a 2 ⫹ 共 a 0 a 1* ⫹a 21a 22兲 , 2 4

共A13兲

f 共 x 兲⫽

⬁

2

⫺⬁

The expression 共B2兲 is proportional to the Fourier convolution of the two functions f (X)⫽(␧ 2 ⫹X 2 ) ⫺1 and g(X) ⫽u(X),

共A14兲

共A15兲

冕 冑 ␲ 1

I共 X 兲⫽

␧ 冑2 ␲ f 䊊g 共 X 兲 . ␲

共B5兲

The convolution operation then gives 共using the table in 关14兴兲

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Im关 I 共 X 兲 ; ␰ 兴 ⫽ ⫽

PHYSICAL REVIEW A 64 053808

where the Hilbert transform H„u ⬘ (X)… is defined by Eq. 共B7兲.

␧ 冑2 ␲ F 共 ␰ 兲 G 共 ␰ 兲 ␲ ␧ 冑2 ␲ ␧ ⫺1 ␲

冑

␲ exp共 ⫺␧ 兩 ␰ 兩 兲 U 共 ␰ 兲 2

APPENDIX C

⫽exp共 ⫺␧ 兩 ␰ 兩 兲 U 共 ␰ 兲 ⫽ 共 1⫺␧ 兩 ␰ 兩 ⫹••• 兲 U 共 ␰ 兲 . 共B6兲 Consider now the Hilbert transform of the function u ⬘ (X) 关15兴 H„u ⬘ 共 X 兲 …⫽

1 ␲

冕

⬁

1 u ⬘ 共 T 兲 dT. ⫺⬁ T⫺X

1 H„u ⬘ 共 X 兲 …⫽ 冑2 ␲ f 䊊g 共 X 兲 . ␲

共B8兲

The convolution operation now gives 共using the table in 关14兴兲 1 Im关 H„u ⬘ 共 X 兲 …; ␰ 兴 ⫽ 冑2 ␲ F 共 ␰ 兲 G 共 ␰ 兲 ␲ 1 冑2 ␲ ␲

冑

Inserting a n ⫽c exp(␮1␪⫹ikn,1␨ ) into Eq. 共C1兲 leads to

␮ 1 ⫽⫺1⫹i v gn k n,1⫺i⍀ n ␴ 1 .

共C2兲

Using now u⫽a n ( ␪ , ␨ )exp(i⍀nT⫹iknx) with T⫽ ␴ ␶ , the growth rate in units of time ␶ is given by 共the terms multiplying ␴ 1 simplify兲 ␭⫽i⍀ n ⫹i v gn ␧k n,1⫺␧.

共C3兲

Now, expanding Eq. 共6兲 with k⫽k n ⫹␧k n,1 , we obtain the same expression as Eq. 共C3兲. If n⫽0, the linearized equation for a 0 is a˙ 0 ⫺iH 共 a 0⬘ 兲 ⫽⫺a 0 ⫺i ␴ 1 a 0 .

␮ 1 ⫽⫺1⫺i 兩 k 0,1兩 ⫺i ␴ 1 . 共B9兲

Comparing Eq. 共B9兲 and the O(␧) correction term in Eq. 共B6兲, we conclude that I 共 X 兲 ⫽u 共 X 兲 ⫺␧H„u ⬘ 共 X 兲 …⫹•••,

共C1兲

共C4兲

Inserting a 0 ⫽c exp(␮1␪⫹ik0,1␨ ) into Eq. 共C4兲 and evaluating H(a 0⬘ ), we find

␲ i sgn共 ␰ 兲共 ⫺i ␰ 兲 U 共 ␰ 兲 2

⫽sgn共 ␰ 兲 ␰ U 共 ␰ 兲 ⫽ 兩 ␰ 兩 U 共 ␰ 兲 .

a˙ n ⫺ v gn a n⬘ ⫽⫺a n ⫺i⍀ n ␴ 1 a n .

共B7兲

Equation 共B7兲 is proportional to the Fourier convolution of and g(X) the two functions f (X)⫽X ⫺1 ⫽u ⬘ (X), i.e.,

⫽

It is interesting to note that the linear terms in Eqs. 共A13兲–共A19兲 could be anticipated from the linear stability analysis. If n⫽0, the linearized problem for a n is

共B10兲

关1兴 Y. Khanin, Principles of Laser Dynamics 共Elsevier, Amsterdam, 1995兲. 关2兴 P. Mandel, Theoretical Problems in Cavity Nonlinear Optics 共Cambridge University Press, Cambridge, 1997兲. 关3兴 C. Szwaj, S. Bielawski, and D. Derozier, Phys. Rev. Lett. 77, 4540 共1996兲. 关4兴 C. Szwaj, S. Bielawski, T. Erneux, and D. Derozier, Phys. Rev. Lett. 80, 3968 共1998兲. 关5兴 J. Miles and D. Henderson, Annu. Rev. Fluid Mech. 22, 143 共1990兲. 关6兴 A. Dinklage, C. Wilke, G. Bonhomme, and A. Atipo, Phys. Rev. E 62, 7219 共2000兲. 关7兴 M. Le Flohic et al., IEEE J. Quantum Electron. 27, 1910 共1991兲. 关8兴 Y. Kuramoto and H. Nakao, Phys. Rev. Lett. 76, 4352 共1996兲. 关9兴 G.F.N. Mazouz, G. Fla¨tgen, and K. Krischer, Phys. Rev. E 55, 2260 共1997兲.

共C5兲

With u⫽a 0 ( ␪ , ␨ )exp(iT) and T⫽ ␴ ␶ , Eq. 共C5兲 implies the growth rate ␭⫽i⫺i␧ 兩 k 0,1兩 ⫺␧,

共C6兲

which is matching Eq. 共6兲 after inserting k⫽␧k 0,1 .

关10兴 A. Gorshkov, G. Lyakhov, K. Voliak, and L. Yarovoi, Physica D 122, 161 共1998兲. 关11兴 In a fiber laser, the homogeneous width decreases with temperature. At room temperature, the homogeneous and the inhomogeneous width have the same order of magnitude (⬇100 Å), thus the system is globally coupled. At liquid nitrogen temperature, the homogeneous width is much smaller (⬇8 Å) than the inhomogeneous width, thus providing a local coupling. 关12兴 G.I. Sivashinsky, Acta Astron. 4, 1177 共1977兲. 关13兴 D. Vaynblat and M. Matalon, SIAM 共Soc. Ind. Appl. Math.兲 J. Appl. Math. 60, 679 共2000兲. 关14兴 I. Gradshteyn and I. Ryzhik, in Table of Integrals, Series and Products 共Academic, New York, 1980兲, p. 1147. 关15兴 D. Zwillinger, in Handbook of Differential Equations 共Academic, New York, 1989兲, p. 254.

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