Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser

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Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser

Herve´ Leblond,¹Mohamed Salhi,¹Ammar Hideur,²Thierry Chartier,²Marc Brunel,²and Franc¸ois Sanchez¹

1Laboratoire POMA, UMR 6136, Universite´ d’Angers, 2 Bd Lavoisier, 49000 Angers, France

2Groupe d’Optique et d’Optronique, CORIA, UMR 6614, Universite´ de Rouen, Avenue de l’Universite´, site Universitaire du Madrillet, Boıˆte Postale 12, 76801 Saint-E´ tienne du Rouvray, France

共Received 20 December 2001; published 13 June 2002兲

We consider a Yb-doped double-clad fiber laser in a unidirectional ring cavity containing a polarizer placed between two half-wave plates. Depending on the orientation of the phase plates, the laser operates in continu- ous, Q-switch, mode-lock, or unstable self-pulsing regime. An experimental study of the stability of the mode-locking regime is realized versus the orientation of the half-wave plates. A model for the stability of self-mode-locking and cw operation is developed starting from two-coupled nonlinear Schro¨dinger equations in a gain medium. The model is reduced to a master equation in which the coefficients are explicitly dependent on the orientation angles of the phase plates. Analytical solutions are given together with their stability versus the angles.

DOI: 10.1103/PhysRevA.65.063811 PACS number共s兲: 42.55.Wd

I. INTRODUCTION

Self-mode-locking has been successfully obtained in both low power and double-clad fiber lasers using additive-pulse mode-locking technique through nonlinear polarization rotation 关1– 6兴. The experimental configuration generally consists in a ring cavity containing a polarizing isolator placed between two polarization controllers. The basic principle is the following. The polarization state evolves nonlinearly during the propagation of the pulse in the fiber due to the combined effects of self-phase modulation and cross-phase modulation induced on the two orthogonal polarization components, both resulting from the optical Kerr effect. A polarization controller is adjusted at the output of the fiber such that the polarizing isolator lets pass the central intense part of the pulse but blocks the low-intensity pulse wings. This technique of nonlinear polarization rotation for passive mode locking leads to stable, self-starting pulse trains. Theoretical approaches to this problem can be divided into two catego- ries. The first one has been proposed by Haus et al.关7兴and consists in writing a phenomenological scalar equation 共the master mode-locking equation兲assuming that all effects per pass are small. The model includes the group-velocity dis- persion共GVD兲, the optical Kerr nonlinearity, and a gain medium. No birefringence is taken into account. Although the mode-locking properties of the laser can be described through this approach for positive and negative GVD, the stability of the mode-locked solutions as a function of the orientation of the polarization controllers cannot be de- scribed in this way. On the other hand Kim et al.关8兴use an approach based on two-coupled nonlinear Schro¨dinger equations. The medium is assumed to be birefringent. The GVD together with the Kerr effect is taken into account. The orientation angles of the eigenaxis of the fiber at both sides of the polarizer are explicitly taken into account through a periodic perturbation. Numerical simulations show that mode locking can be achieved for negative GVD. To the best of our knowledge, none of the models take into account all the

characteristics of the fiber共birefringence, GVD, gain, optical Kerr effect兲and the action of the orientation of the polarization controllers.

The aim of this paper is to investigate experimentally and theoretically the mode-locking properties of the Yb-doped double-clad fiber laser. Mode locking is achieved through nonlinear polarization rotation in a unidirectional ring cavity containing a polarizer placed between two half-wave plates.

This paper is organized as follows. Section II is devoted to the experimental results. We first briefly recall the operating regime of the laser as a function of the orientation of the two half-wave plates. A stability diagram of the mode-locked laser is presented关9兴. The parameters required in the modeling section are measured. The birefringence of the double-clad fiber is first determined using the magneto-optic method 关10兴. The total dispersion has been estimated from other experiments where compression of the pulses was realized 关11,12兴. In Sec. III the stability of the mode-locking regime is theoretically investigated. The analysis is based on the coupling between the two linearly polarized eigenstates of the fiber关8,13兴. The propagation problem is modeled through two nonlinear Schro¨dinger equations in a gain medium, where the GVD, the nonlinear coefficient, and the gain are assumed small over the cavity length. The eigenaxes of the fiber at each side of the polarizer are referenced with respect to the polarizing axis. These angles can be adjusted with two half-wave plates and will thus be considered as variable in the analysis. The polarizer is treated as a periodic perturbation acting as a projecting operator. Using the expressions derived in the propagation problem, we calculate the electric- field amplitude just after the polarizer as a function of the corresponding amplitude at the previous round trip. The round-trip number is then considered as a continuous variable, which allows one to use a multiscale analysis approach.

The final equation is a complex Ginzburg-Landau equation whose coefficients explicitly involve the orientation angles of the half-wave plates. Explicit solutions are given and their stabilities are discussed. A stability diagram of the mode-

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locked regime is calculated and compared to the experimental diagram.

II. EXPERIMENTAL RESULTS

In this section we briefly recall the results obtained with an Yb-doped double-clad fiber laser in a unidirectional ring cavity operating at 1.08 ␮^m 关9兴. A polarizing isolator is placed between two half-wave plates. Although this configuration is known to generate ultrashort pulses in low-power fiber lasers, it can also lead to very different regimes depending on the orientation of the polarization elements, such as cw operation, Q-switching, mode-locking, or unstable emis- sion regime. We thus perform a systematic study of the operating regime as a function of the relative orientation of the two half-wave plates. The experimental setup is shown in Fig. 1. The 4-m-long Yb-doped double-clad fiber is side pumped using thev-groove technique关3,14兴. The launching efficiency is about 70%. This pumping scheme is very convenient for ring laser applications because the two fiber ends remain free. The fiber core diameter is 7 ␮m, allowing a single-mode laser signal propagation. The inner cladding is a 125⫻125 ␮^m² square leading to a multimode pump propagation. The geometry of the inner cladding ensures an effi- cient pump absorption. Indeed, the pump power is almost completely absorbed along the doped fiber. In addition, multimode propagation is required for high injection efficiency of large area laser diodes. Two single-mode fibers at 1 ␮^m are spliced at both ends of the double-clad fiber leading to a total fiber length of about 8.5 m, the cavity length is about 9 m. Fiber ends are angle cleaved in order to avoid any Fresnel reflection that could generate a standing-wave operating regime. The splicing losses are less than 0.1 dB. The doped fiber is pumped at 975 nm with a 4-W-semiconductor laser. A bulk polarizing optical isolator is used to obtain a traveling wave laser. The latter is placed between two half-wave plates at 1.08 ␮m. Each plate can be rotated in a plane perpendicu- lar to the propagation direction. A partially reflecting mirror ensures an output coupling of about 90%. The output beam is analyzed through different means. A fast photodiode is used when the laser is cw, Q switched, or unstable. In the regular mode-locking regime, we use an optical autocorrelator to measure the duration of the pulses. Note that there is no compression system in the cavity. An optical spectrum ana-

lyzer 共Advantest兲together with a microwave spectrum ana- lyzer共Tektronix兲are also used when needed. Finally, a pow- ermeter allows us to measure the average output power.

We investigate now the operating regime of the laser as a function of the orientation of the two waveplates.␪1 (␪2) is the orientation of one eigenaxis of the first 共second兲 half- wave plate referenced to the input 共output兲polarizer of the optical isolator. The angle between the input and output polarizer of the isolator is 45 deg. For a given pumping power we record the evolution of the output intensity versus␪1and

␪2. Practically, we fix ␪1 and measure the particular values of␪2for which a change in the operating regime occurs. It is convenient to represent these results in the plane (␪1,␪2). It allows to easily identify the output regime of the laser. The experimental results, obtained for a pumping power of about 2.75 W, are summarized in Fig. 2. We have experimentally observed that the different output regimes are periodic versus

␪1 and␪2 with a period of 90 deg. Figure 2 shows that the laser essentially operates in cw, mode-lock, or unstable re- gime while the Q-switch behavior appears near the boundary between cw and mode-lock regimes. Note that since the operating regime depends on the pumping rate, the mapping of Fig. 2 is pump power dependent. Typically, when the pump power decreases 共increases兲, the ranges of mode-locking decrease 共increase兲.

The different dynamical behaviors observed in our experiment have been discussed in detail in Ref. 关9兴. Here, we concentrate our attention on the mode-locked operating regime. In Fig. 3 we give the results obtained for ␪1

⫽20 deg and␪2⫽0 deg. In this case, the laser is first cw just above threshold 关Fig. 3共a兲兴. While the pump power is increased, an irregular self-pulsing regime appears, which finally becomes a regular self-mode-locking regime. As previously discussed in the Introduction, the basic principle responsible for the mode locking is well known in such configurations. It is based on Kerr effect through nonlinear polarization rotation. This technique is also called FIG. 1. Experimental setup.

FIG. 2. Operating regime of the laser as a function of the orientation of the two half-wave plates␪1 and␪2. The pump power was 2.75 W. ML stands for mode-locked regime and cw for continuous wave regime.

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polarization-additive pulse mode locking. The repetition rate of the pulses corresponds to the free spectral range of the cavity, about 20 MHz in our case. Autocorrelation traces showed a coherence spike of about 150 fs on a longer ped- estal of about 60 ps 关6兴. A typical spectrum in the mode- locking range is given in Fig. 3共b兲. The mode-locked spectrum bandwidth is more than 25 nm. This spectrum suggests that shorter pulses can be achieved by a suitable dispersion compensation. We have recently shown that femtosecond pulses are obtained with a grating pair inserted in the cavity 关6兴. In the present mode-locked regime, we measure 600 mW output power for 3.25 W pump power, which corresponds to high-energy pulses of about 30 nJ.

Consider now the measurements of parameters required in the theoretical section. We first measure the beating length L_B of the double-clad fiber using the magneto-optic method 关10兴. The method consists in applying a sinusoidal magnetic field produced by a coil in a small portion of the fiber and to move the coil along the fiber axis. The fiber is placed between an analyzer and a polarizer. This system is then illu- minated and analyzed in transmission. The detected output power is a periodic signal versus the coil displacement. The period in the spatial domain coincides with the beating length of the fiber. We found L_B⫽50 cm leading to a bire- fringence parameter K⫽1/(L_B)⫽2 m^⫺¹. We must also consider the birefringence of the undoped fibers. It has been measured to be of about 1 m^⫺¹. In the following we will consider an average value of the birefringence of about 1.5 m^⫺¹. A second important parameter is the GVD. It has been estimated from compression experiments 关12兴. We find ␤2

⫽0.026 ps²/m.

III. THEORY

The aim of this section is to develop a theoretical model able to describe the mode-locking properties of the Yb-doped fiber laser as a function of the orientation of the two half- wave plates. For that, we consider a birefringent fiber in a unidirectional ring cavity with an intracavity polarizer. The orientations of the eigenaxes of the fiber at each side of the polarizer are unspecified and will be taken as variable parameter. This is equivalent to considering that the eigenaxes of the fiber are fixed and two half-wave plates are used to adjust their orientations with respect to the polarizer.

A. Derivation of a master equation 1. Propagation in the fiber

The starting point is the equations giving the evolution of the two polarization components in a gain medium with Kerr nonlinearity and GVD. In the framework of the eigenaxis of the birefringent fiber moving at the group velocity, the pulse envelope evolution is described by the following system 关8,13兴:

i⳵zu⫺Ku⫺␤2

2 ⳵t

2u⫹␥共u兩u兩²⫹Au兩v兩²⫹Bv²u*_兲

⫽ig

冉

¹^⫹^␻¹^g²^⳵^t²

冊

^u, ^共¹^兲

i⳵zv⫹Kv⫺␤2

2 ⳵t

2v⫹␥共v兩v兩²⫹Av兩u兩²⫹Bu²v*_兲

⫽ig

冉

¹^⫹^␻¹^g²^⳵^t²

冊

^v, ^共²^兲

where⳵tdenotes the partial derivative operator ⳵^/⳵^t. ␤2 共in ps²/m) is the GVD coefficient; K共in m^⫺¹) is the birefringent parameter and is related to the x and y refractive indexes through the relation K⫽␲⁽ⁿx⫺n_y)/␭, where lambda is the optical wavelength;␥⫽2␲ⁿ2/(␭A_{e f f}) is the nonlinear coef- ficient; n₂ 共in m²/W) is the nonlinear index coefficient; A_{e f f} 共in m²) is the effective core area of the fiber; and A and B are the dielectric coefficients. In isotropic media, A⫽2/3 and B

⫽1/3 关13兴. This is the case with our silicate fibers. g is the gain coefficient 共in m^⫺¹) and ␻g is the spectral gain band- width共in ps^⫺¹). At first order the gain coefficient is fixed by the fact that it compensates the losses. Note that polarization mode dispersion is not taken into account since the cavity length is short 共about 9 m兲. For numerical simulations we will use the following values for the parameters: K⫽1.5 m^⫺¹, ␤2⫽0.026 ps²/m, L⫽9 m, ␥⫽3⫻10^⫺³ W^⫺¹ m^⫺¹, and␻g⫽10¹³s^⫺¹.

We now assume that the GVD ␤2, the nonlinear coefficient␥, the gain filtering␳⫽g/␻g

2 are small over one round trip of the cavity. We introduce a small parameter ␧ and replace these quantities by ␧␤2, ␧␥^{, and}␧␳. We then look for solutions of the set of equations共1兲–共2兲under the form of a power series in␧, as

FIG. 3.␪1⫽20 deg and␪2⫽0 deg.共a兲Laser characteristic and 共b兲typical optical spectrum in the mode-locked regime.

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u⫽u₀⫹␧u₁⫹O共␧²兲, 共3兲 v⫽v0⫹␧v1⫹O共␧²兲. 共4兲 At order␧⁰ it yields

u₀⫽u˜₀e^(g^⫺^iK)z, 共5兲 v₀⫽v˜

0e^(g^⫹^iK)z, 共6兲

where u˜

0and˜v

0 do not depend on z, i.e., depend on the time variable t only.

Making the transformation

u₁⫽˜u₁共z,t兲e^(g^⫺^iK)z, 共7兲 v₁⫽˜v

1共z,t兲e^(g^⫹^iK)z, 共8兲 the equations obtained at order ␧ can be integrated with re- gard to z to yield, for the component u,

˜u

1⫽u˜

1共0兲⫹z

冉

^␳^⫺ⁱ^␤²²

冊

^⳵^t²^˜û⁰^⫹ⁱ^␥^共û^˜⁰^兩^˜û⁰^兩²^⫹Âu^˜⁰^兩^˜^v⁰^兩²^兲

⫻e^2gz⫺1 2g ⫹i␥Bv˜

0 2u˜

0*^e

(2g⫹4iK)z⫺1

2g⫹4iK . 共9兲

To get the complete expression for u(z), we notice that

u共0兲⫽˜u₀⫹␧˜u₁共0兲⫹O共␧²兲. 共10兲

Then the two components of the wave amplitude can be written as

u共z兲⫽u共0兲e^(g^⫺^iK)z⫹␧

冋

^z

冉

^␳^⫺ⁱ^␤²²

冊

^⳵^t²^u^共⁰^兲

⫹i␥关u共0兲兩u共0兲兩²⫹Au共0兲兩v共0兲兩²兴e^2gz⫺1 2g

⫹i␥^B^˜^v共0兲²u˜共0兲*^e

(2g⫹4iK)z⫺1

2g⫹4iK

册

ê^(g^⫺îK)z^⫹Ô^共␧²^兲^,

共11兲

v共z兲⫽v共0兲e^(g^⫹^iK)z⫹␧

冋

^z

冉

^␳^⫺ⁱ^␤²²

冊

^⳵^t²^v^共⁰^兲

⫹i␥关v共0兲兩v共0兲兩²⫹Av共0兲兩u共0兲兩²兴e^2gz⫺1 2g

⫹i␥^Bu^˜共0兲²v˜共0兲*^e

(2g⫺4iK)z⫺1

2g⫺4iK

册

ê^(g^⫹îK)z^⫹Ô^共␧²^兲^.

共12兲

2. Polarizer

Figure 4 shows the respective orientations of the eigenaxes of the fiber before and after the polarizer 共optical

isolator兲. (u_⫺,v_⫺) are the field components just before the polarizer, (u_⫹,v_⫹) just after it. The effect of the polarizer can be written as

冉

^v^u^⫹⫹

冊

^⫽^␤

冉

^cos^sin^␪^␪⫹^⫹

冊

^共^cos^␪^⫺^sin^␪^⫺^兲

冉

^v^u^⫺⫺

冊

^, ^共¹³^兲

where␤ is its transmission coefficient (␤⬍1). Immediately after the polarizer, the field has a well-defined linear polar- ization. We denote by f_nits amplitude at the beginning of the nth round trip. Equation共12兲can be written as

f_n_⫹₁⫽␤共cos␪_⫺^sin␪_⫺兲

冉

^u^v^⫺⫺^,n,n

冊

^共¹⁴^兲

and

冉

^u^v^⫹⫹^,n,n^⫹⫹¹1

冊

^⫽

冉

^cos^sin^␪^␪⫹^⫹

冊

^fⁿ^⫹¹^, ^共¹⁵^兲

where (u_n,vn) are the field components during the nth round. The field components u_⫹_,n⫽u_n(0) andv_⫹_,n⫽v_n(0), at the entrance of the fiber immediately after the polarizer, are transformed, respectively, into u_n(L)⫽u_⫺_,n and vn(L)

⫽v_⫺_,n, at the end of the fiber immediately before the polar- izer (L is the fiber length兲.

u_n(0) andv_n(0) are proportional to f_n. Using the propagation formulas共11兲and共12兲, u_n(L) andvn(L) can be com- puted. Then f_n_⫹₁ is computed as a function of f_n using Eq.

共14兲. This yields

f_n_⫹₁⫽␤^e^gL

再

^{Q f}ⁿ^⫹^␧

冋冉

^␳^⫺ⁱ^␤²²

冊

^LQ^⳵^t²^fⁿ^⫹^{i P f}ⁿ^兩^fⁿ^兩²

册冎

⫹O共␧²兲, 共16兲 with

Q⫽关cos共␪_⫹⫺␪_⫺兲cos KL⫺i cos共␪_⫹⫹␪_⫺兲sin KL兴, 共17兲 and

FIG. 4. Definition of the angles␪⫺and␪⫹.

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P⫽␥

冉

^e^2gL^2g^⫺¹

冋

^cos^共^␪^⫹^⫺^␪^⫺^兲^{cos KL}^⫺^{i cos}^共^␪^⫹^⫹^␪^⫺^兲

⫻sin KL⫹A⫺1

2 sin 2␪_⫹关sin共␪_⫹⫹␪_⫺兲cos KL

⫺i sin共␪_⫹⫺␪_⫺兲sin KL兴

册

⫹B

2 sin 2␪⫹

冋

^sin^␪^⫹^cos^␪^⫺ê^⫺îKLê^(2g^2g^⫹^⫹^4iK)L^4iK^⫺¹

⫹cos␪_⫹^sin␪_⫺êîKLê

(2g⫺4iK)L⫺1

2g⫺4iK

册冊

^. ^共¹⁸^兲

3. Gain threshold and continuous limit for f_n

The dominant behavior of ( f_n) is given by the zero-order term in Eq. 共16兲:

f_n_⫹₁⫽␤^e^gL^{Q f}n⫹O共␧兲. 共19兲

A steady state over a large number of round trips could be reached if this first-order evolution is a multiplication by a phase factor, but only in this case. Therefore the modulus of

␤^e^gLQ must be 1. This gives the threshold gain value g₀⫽⫺1

2L ln共␤²兩Q兩²兲

⫽⫺1

2L ln兵␤²关cos²共␪_⫹⫺␪_⫺兲⫺sin 2␪_⫹^{sin 2}␪_⫺^sin²^KL兴其^. 共20兲 Denote by eⁱ^␺ the quantity ␤^e^g⁰^LQ. The exact condition is 兩␤^e^gL^Q兩⫽1⫹O(␧), the gain g, therefore, writes g⫽g₀

⫹␧g₁, in which g₁ is still free. Expanding e^␧^g¹^Lin a power series of␧, Eq.共16兲becomes

f_n_⫹₁⫽eⁱ^␺共1⫹␧g₁L兲f_n⫹␧

冉

^␳^⫺ⁱ^␤²²

冊

^Leⁱ^␺^⳵^t²^fⁿ

⫹i␧Pe^i␺

Q f_n兩f_n兩²⫹O共␧²兲. 共21兲

Consider now some function f of a continuous variable z, obeying an equation of the form

i⳵zf⫽共A⫹i␧B兲f⫹␧C⳵t

2f⫹␧Df兩f兩², 共22兲

whereAandBare real, and CandDcomplex coefficients.

We look for solutions under the form f⫽f₀⫹␧f₁⫹•••. The solution at the first order is

f₀⫽f˜₀e^⫺ⁱ^A^z. 共23兲 We write

f₁⫽f˜₁共z,t兲e^⫺ⁱ^A^z, 共24兲 and obtain

f共L兲⫽关f共0兲⫹␧兵Bf共0兲⫺iC⳵t 2f共0兲

⫺iDf共0兲兩f共0兲兩²其^L^兴ê^⫺ⁱÂ^L^⫹Ô^共␧²^兲^. ^共²⁵^兲 Equations 共21兲 and 共25兲 are identified according to f (0)

⬅f_n and f (L)⬅f_n_⫹₁,to yield

e^⫺ⁱ^A^L⫽eⁱ^␺, 共26兲 B⫽g₁, 共27兲 C⫽␤2

2 ⫹i␳^, 共28兲

D⫽⫺P

QL. 共29兲

Then the continuous equation 共22兲yields some interpolation of the discrete sequence ( f_n). The continuous approximation is relevant when the number of round trips is very large;

further, mode-locked pulses actually correspond to this situ- ation. The small correction of order ␧ in equations 共22兲 or 共25兲 gives account for the variations of f on ‘‘propagation distances,’’ which are here number of round trips, very large, of order 1/␧. This can be shown in a rigorous way using the multiscale formalism, commonly used for the derivation of the model equations in the soliton theory关15兴. We introduce a slow variable ␨⫽␧z, in such a way that

⳵z⫽⳵zˆ⫹␧⳵_␨^. 共30兲 The values of␨ about 1 correspond to number of round trips about 1/␧. f is expanded as above as f₀⫹␧f₁⫹¯, and the first order obviously yields Eq.共23兲, we make the transform 共24兲, and f˜₁ must satisfy the following equation:

i⳵␨f˜₀⫹i⳵zˆf˜₁⫽iBf˜₀⫹C⳵t

2f˜₀⫹Df˜₀兩f˜₀兩², 共31兲

Equation共31兲can be written as

⳵zˆf˜₁⫽F共f˜₀兲. 共32兲

The long-distance evolution of f˜₀ is obtained from the re- quirement that f˜₁ does not grow linearly with z. This yields

⳵zˆf˜₁⫽0 and

i⳵␨f˜₀⫽iBf˜₀⫹C⳵t

2f˜₀⫹Df˜₀兩f˜₀兩². 共33兲 or, using real coefficients only,

i⳵␨f˜₀⫽ig₁f˜₀⫹

冉

^␤²²^⫹ⁱ^␳

冊

^⳵^t²^f˜⁰^⫹^共D^r^⫹ⁱ^Dⁱ^兲^f˜⁰^兩^f˜⁰^兩²^. ^共³⁴^兲

Equation共33兲or共34兲is the cubic complex Ginzburg-Landau 共CGL兲 equation. Note that Eq. 共33兲 is formally identical to the master equation proposed by Haus et al. 关7兴. However, the coefficients of Eq.共33兲explicitly depend on the orientation of the eigenaxis of the fiber at both sides of the polarizer.

An essential feature is the arising of a nonlinear gain or absorption Di, which results from the combined effects of

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the nonlinear rotation of the polarization, the losses due to the polarizer, and the linear gain. The value and the sign of Didepend on the angles␪⫹and␪⫺between the eigenaxis of the fiber and the polarizer.

B. Different regimes for the CGL equation 1. The stationary solution and its modulational instability The CGL Eq.共34兲 admits the following nonzero stationary solutions, i.e., with a constant modulus:

f₁⫽Aeⁱ⁽^␬␨⫺⍀^t) 共35兲 with

⍀²⫽1

␳^共^Dⁱ^兩^A^兩²^⫹^g¹^兲 ^共³⁶^兲 and

␬⫽␤2

2␳^共Dⁱ^兩^A^兩²^⫹^g¹^兲⫺^D^r^兩^A^兩²^. ^共³⁷^兲

In the particular case ⍀⫽0 it is a constant solution 共inde- pendent of t), with a fixed amplitude

A⫽

冑

^⫺D^gi¹

, 共38兲

and

␬⫽Dr

Di

g₁. 共39兲

It exists only ifDig₁⬍0, i.e., if the excess of linear gain g₁ and the effective nonlinear gainDihave opposite signs. We perform a linear stability analysis for the constant solution f_c⫽Aeⁱ^␬␨. We seek for solutions of the form f₁⫽f_c(1

⫹u), u being very small. It is seen that it must satisfy the following equation:

⫺␬^u⫹i⳵␨u⫽ig₁u⫹

冉

^␤²²^⫹ⁱ^␳

冊

^⳵^t²û^⫹^DÂ²^共^2u^⫹û^*^兲^.

共40兲 Due to the term u*, Eq. 共40兲 is not linear. The eventual instability is of the type discovered first by Benjamin and Feir in the frame of water waves关16兴. It has been shown that this kind of instability involves the nonlinear interaction of two modulation terms with conjugated phases with the square A² of the constant solution关17兴. Thus we write u as

u⫽u₁e⁽^␭␨⫺ⁱ^␻^t)⫹u₂e⁽^␭^*^␨⫹ⁱ^␻^t). 共41兲 It is found that ␭ satisfies an equation of the form

␭²⫹2b␭⫹c⫽0. 共42兲

Using the expression 共38兲 of the fixed amplitude A, we get the following values of the real constants b and c:

b⫽g₁⫹␳␻²^, 共43兲

c⫽

冉

^g¹^D^D^rⁱ^⫹^␤²²^␻²

冊

²^⫹^b²^⫺^g¹²

冉

¹^⫹^D^D^r²ⁱ²

冊

^. ^共⁴⁴^兲

A short analysis of Eq.共42兲show that a necessary condition for u to remain bounded is that b is positive. This proves that modulational instability occurs as soon as the excess of lin- ear gain g₁ is negative and the nonlinear gainDiis positive.

Recall indeed that the existence of the constant nonzero solution requires that these two quantities have opposite signs.

When g₁⬎0 andDi⬍0, we show that the modulational instability never occurs. Numerical computation shows indeed that the quantity (Dr␤2/2⫹Di␳) is always negative, which ensures, using elementary analysis, that c is always positive, and the result follows.

If we admit that the excess of linear gain g₁ will self- adjust to a value for which a stable solution exists, the above discussion shows that the stability of the constant solution depends on the sign of the effective nonlinear gainDi. It is stable only when Di⬍0. We expect to observe continuous laser emission only when the constant solution is stable.

Thus the domains of continuous emission should coincide with the regions where Di is negative. The sign of Di as function of the angles (⫺1/2␪_⫺^,1/2␪_⫹) is drawn in Fig. 5. It is easily checked that the relation between the angles

␪1, ␪2 used in the experiments and the angles ␪_⫹^, ␪_⫺ defined by Fig. 4 is such that ␪1⫽⫺1/2␪_⫺⫹␪10 and ␪2

⫽1/2␪_⫹⫹␪20, with some fixed value of ␪10 and␪20. The white domain in Fig. 5 corresponds to the negative values of the nonlinear gainDi. It is thus the domain where the constant solution of the CGL equation is stable. Continuous laser emission should occur in this domain.

2. Localized solutions

A localized analytical solution of the CGL Eq. 共34兲 can also be written关18兴. It has the following expression:

f₁⫽a共t兲e^i(dlna(t)^⫺␻␨⁾, 共45兲 FIG. 5. Sign of the effective nonlinear gainDias a function of the orientation of the polarizer. White:Di⬍0, and the constant solution of the CGL equation is stable. Gray:Di⬎0, and the constant solution of the CGL equation is unstable.

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with

a共t兲⫽BC sech共Bt兲共46兲

and

B⫽

冑

␳^d²⫺^g␳¹⫺␤2d, 共47兲 C⫽

冑

²^3d共␤^共2⁴D^␳i²⫺^⫹2^␤␳²²D^兲r

, 共48兲

d⫽3共␤2Dr⫹2␳Di兲⫹

冑

⁹共2␳Di⫹␤2Dr兲²⫹8共␤2Di⫺2␳Dr兲^e

2共␤2Di⫺2␳Dr兲 , 共49兲

␻⫽⫺g₁共4␳^d⫹␤2d²⫺␤2兲

2共␳^d²⫺␳⫺␤2d兲 . 共50兲

The inverse B of the pulse length is real only if the quantity T⫽(␳^d²⫺␳⫺␤2d) and the excess of linear gain g₁have the same sign. The regions whereT is either positive or negative are specified in Fig. 6. The background with zero amplitude is stable when the excess of linear gain g₁is negative, and is unstable otherwise. When g₁⬎0, the exact localized solution 共45兲is unstable due to the instability of the background关18兴. Qualitatively, localized pulse formation can be expected when the excess of linear gain g₁is negative and the nonlinear gain Di is positive, as is suggested in Fig. 7. The effective self-phase modulation Dr is always negative, and the dispersion ␤2 is positive. Therefore their conjugated effect leads to the increase of the pulse width. If g₁is positive and Dinegative关Fig. 7共a兲兴, the nonconservative effects decrease the amplitude at the top of the pulse, and increase it at the bottom: no stable localized pulse can be formed. If on the contrary g₁is negative andDi⬎0关Fig. 7共b兲兴, energy appears at the top of the pulse and disappears at its bottom, yielding some pulse narrowing, which could be expected to balance the broadening caused by the nonlinear index variations. In fact the analytical localized solution 共45兲 is never stable when␤2Dr⬍0, but can be stabilized when higher-order non-

linear terms are taken into account, as shows the study of the so-called quintic CGL equation 关18兴. The stabilizing term should be a quintic nonlinear absorption. Observe that here the effective nonlinear gain Di follows from the linear gain and the nonlinear evolution of the polarization in the fiber, without considering cubic nonlinear gain in the initial models 共1兲 and 共2兲. In an analogous way, an effective quintic nonlinear gain term in the master equation 共33兲 does not require that a quintic nonlinear gain term is present in the initial model, either a cubic nonlinear gain or a quintic nonlinear index should be enough. The precise determination of this higher-order term is left for further study. On these grounds, the exact solution共45兲corresponds to a potentially stable pulse when g₁⬍0 and Di⬎0, and to a completely unstable pulse when g₁⬎0 andDi⬍0.

3. The different regimes

According to the conclusions of the preceding section, and assuming that the excess of linear gain g₁will self-adjust to the value for which a stable solution exists, mode-locked laser emission can be expected in the region where Di⬎0 andT⬍0. This is the domain in the dark gray marked ML in Fig. 8. In the region whereDiis negative, the constant solution of the CGL equation is stable, and continuous laser emission is expected to occur. This is the white domain marked cw in Fig. 8. In the region where T is positive and Di negative, no stable localized or constant solution exists.

The laser behavior is expected to be unstable in this region, as is shown in light gray in the Fig. 8. The theoretical results summarized in the Fig. 8 can be compared to the experimental results of Fig. 2. A discrepancy between the two figures appears at first glance: the periodicity of the behavior regard-

FIG. 6. Sign of the excess of nonlinear gain value g₁for which the exact localized solution exists, as a function of the orientation of the polarizer. White:Tand g₁positive. Gray:Tand g₁negative.

FIG. 7. Schematic representation of the effect of the nonlinear gainDi, of the excess of linear gain g₁, and of both the dispersion

␤2and the effective self-phase modulationDron a localized pulse.

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ing the variable␪2or␪_⫹/2 seems to be 45 deg according to the theory, while it is 90 deg according to the experiment.

Indeed, the two elongated domains of mode-locked behavior are not equivalent according to experimental observation, while they appear to be identical on the theoretical Fig. 8. A contour plot of the effective nonlinear gainDi as a function of the angles is drawn in Fig. 9. It is seen that the values of Diare not the same in the two theoretical domains of mode- locked emission. Thus they are in fact not equivalent, and the periodicity of the theoretical result is 90 deg. Further it has been shown 关18兴 that an excessive value of the nonlinear gain might prevent pulse stabilization. A more accurate analysis would thus very likely show that the regions where Ditakes its highest values are out of the domain of stability of the localized pulse. This confirms that the two horizontal domains of the mode-locked regime drawn in Fig. 8 are not equivalent. Further, it could explain a part of the discrepancy between theoretical and experimental results.

Before we conclude, let us consider the influence of the group velocity dispersion on the stability of the mode- locking solutions. This is important because GVD compensation is required to generate subpicosecond pulses. Experi- mentally, we have obtain 666 fs pulses with a grating pair inserted in the cavity关6,12兴. Under these conditions, the total GVD is about ␤2⫽0.0005 ps²/m. Hence, stable ultrashort pulse generation is possible for such low values of the GVD.

We have tested our master equation for decreasing values of

␤2. We have found that a lowering of␤2 results in a reduc- tion of the stable mode locking regions. In addition, mode- locking completely disappears for␤2ⱗ0.0015 ps²/m. These results are in agreement with the predictions of the Haus’s model关7兴but not with our experimental results. The validity of the model presented here is therefore limited to GVD values not too close to zero. Higher-order terms of the GVD could be required to improve the model near the zero- dispersion point. This problem deserves further experimental and theoretical work.

IV. CONCLUSION

We have given a rigorous derivation of the master equation, a CGL equation, that allows one to discuss theoretically the different operating regimes of the passively mode-locked Yb-doped double-clad fiber laser. This equation involves a single amplitude, but gives an account of the evolution of the polarization inside the fiber through its coefficients. An effective nonlinear gain arises from the conjugated effects of the linear gain and the self-phase modulation. Its sign depends on the orientation of the polarizer. The existence and stability of the constant nonzero solution of the CGL equation have been discussed. It gives an account of the continuous emission regime of the laser. An analytical localized solution has also been given. Although it is properly speaking unstable, it allows one to discuss the formation of localized pulses, i.e., the mode-locked regime of the laser. An unstable regime has also been identified. These theoretical results are in good accordance with the experiments. The study of higher-order corrections to this model should improve these results.

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FIG. 8. Operating regime of the laser as a function of the angles

␪⫹ and ␪⫺ between the polarizer and the eigenaxis of the fiber, according to the theory.

FIG. 9. Contour plot of the effective nonlinear gain Di as a function of the angles ␪⫹ and ␪⫺ between the polarizer and the eigenaxis of the fiber.

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