DOI: 10.1007/s00340-006-2214-1

Appl. Phys. B 83, 503–510 (2006)

Lasers and OpticsApplied Physics B

v.l. kalashnikov1,

e. podivilov2

a. chernykh2

a. apolonski2,3

Chirped-pulse oscillators:theory and experiment1 Institut für Photonik, TU Wien, Gusshausstr. 27/387, 1040 Vienna, Austria2 Institute of Automation and Electrometry, RAS, 630090 Novosibirsk, Russia3 Department für Physik der Ludwig-Maximilians-Universität München,

Am Coulombwall 1, 85748 Garching, Germany

Received: 18 January 2006/Revised version: 16 March 2006Published online: 14 April 2006 • © Springer-Verlag 2006

ABSTRACT Theory of chirped-pulse oscillators operating in thepositive dispersion regime is presented. It is found that thechirped pulses can be described analytically as solitary pulsesolutions of the nonlinear cubic-quintic complex Ginzburg–Landau equation. Due to the closed form of the solution, basiccharacteristics of the regime under consideration are easilytraceable. Numerical simulations validate the analytical tech-nique and the chirped-pulse stability. Experiments with 10 MHzTi:Sa oscillator providing up to 150 nJ chirped pulses, which arecompressible down to 30 fs, are in agreement with the theory.

PACS 42.65.Re; 42.65.Tg; 42.55.Rz

1 Introduction

Oscillators directly generating femtosecond pulseswith energy exceeding 100 nJ are of interest for numer-ous applications, such as frequency conversion, frequencycomb generation, micro-machining, etc. A well-known wayto achieve the over-µJ femtosecond pulse energy is realizedby oscillator-amplifier systems. However, such systems havehigh cost and comparatively low energy stability and pulserepetition rate. On the other hand, only ≈ 10 nJ femtosecondpulses are achievable directly from a femtosecond solid-stateoscillator operating in the Kerr-lens mode-locking (KLM)regime at 100 MHz pulse repetition rate (for sub-50 fs pulses).Cavity dumping allows increasing the pulse energy from suchan oscillator up to 100 nJ [1, 2], but makes the system morecomplex.

A promising approach to the µJ pulse energy frontier isbased on an considerable decrease of the oscillator repeti-tion rate [3, 4]. However, such long-cavity oscillators sufferfrom strong instabilities caused by enhanced nonlinear effectswithin an active medium, which result from high intracavitypower. It is possible to suppress instabilities by means of thepulse power lowering due to the pulse stretching. If the pulseis a soliton [5], such stretching requires a fair amount of thenet negative group-delay dispersion (GDD; all abbreviationsare summarized in Table 1) inside the oscillator cavity [6, 7].This increases the pulse width, which cannot be reduced fur-

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ther because the formed soliton is chirp-free. An alternativetechnique is to use an oscillator operating in the positive dis-persion regime (PDR) [8–11]. In PDR the chirped solitarypulse (CSP) develops. On the one hand, owing to its picosec-ond width, CSP has the peak power lower than the criticalpower of self-focusing Pcr inside an active medium. This re-duces the pulse instability substantially. On the other hand,owing to its huge chirp, CSP is compressible down to sub-30 fs width [11]. As a result, the achievable peak powers ex-ceed 10 MW and, after focusing, the peak intensity can exceed1015 W/cm2. This opens a way to low-cost, high-stable andcompact light sources for highly-nonlinear physics, materialprocessing, etc.

PDR positive dispersion regimeCSP chirped solitary pulseGDD group-delay dispersionKLM Kerr-lens mode-lockingSPM self-phase modulationSAM self-amplitude modulationCGLE complex Ginzburg–Landau equationβ GDD coefficientα square of the inverse gain bandwidthκ SAM parameterζ parameter of SAM saturationγ SPM coefficientc ≡ αγ/βκ control parameter defining PDRσ net-loss coefficientA slowly varying field amplitudeE spectral amplitudeP(t) (P(0)) intracavity instant (peak) powerPcr critical power of self-focusingPav (Pout

av ) intracavity (output) cw power (i.e. averaged power)p spectral powerE (Eout) intracavity (output) pulse energyE∗ energy stored inside the oscillator cavity in cw regimeδ “stiffness coefficient” of the gain saturationT intracavity pulse durationTcav cavity periodTr gain relaxation timeω frequencyΩ instant frequency∆ parameter of the spectral truncationΩL half-width of the Lorentzian spectral profileψ (Q) chirp (spectral chirp)χ output mirror transmission coefficientg (l) gain (loss) coefficient

TABLE 1 Basic abbreviations and symbols (primed symbols correspondto normalized values)

504 Applied Physics B – Lasers and Optics

CSP is not allied to the chirp-free Schrödinger soliton be-cause the last can be formed only by the strict balance betweennegative GDD and self-phase modulation (SPM) inside activemedium or fiber. Therefore PDR is a challenge for the the-oretical analysis and requires taking into account the varietyof linear and nonlinear processes involved into the pulse for-mation. There are two approaches providing an insight intonature of CSP. The first one is based on the methodology de-veloped for fiber oscillators, where GDD can vary along thepropagation axis. In this case, a so-called self-similar pulsecan be formed [12, 13]. Such a pulse is heavily chirped andhas a parabolic-like spectrum with truncated edges. Anotherapproach considers CSP as a solitary wave formed under con-dition of positive GDD, whose variation along the propaga-tion axis does not play any essential role in the pulse forma-tion [14, 15]. The last approach describes a variety of spectra,which are typical for CSP, and has represented quite accuratedescription of the experiments with Ti:sapphire oscillators op-erating in PDR [11, 14].

In this paper we shall present an advanced theory of PDRtreating CSP as a solitary wave. This theory is based onthe generalized cubic-quintic nonlinear complex Ginzburg–Landau equation (CGLE) [16] describing KLM in a stronglynonlinear regime. The approximate but accurate analytical so-lution for CSP is reported, which allows tracing the regimecharacteristics in a broad range of the oscillator parameters.We show that the numerical experiments prove the stability ofCSP in PDR and give an excellent agreement with the analyti-cally obtained pulse characteristics (such as pulse and spectralwidths, spectral profile, energy, etc.). We demonstrate that ourtheoretical results are in good concordance with the experi-mental data for high-energy Ti:sapphire oscillators operatingin PDR.

2 Chirped solitary pulses in PDR as stationarysolutions of the cubic-quintic nonlinear CGLEKLM is based on a spatial nonlinear effect, viz self-

focusing, providing additional gain for intense components ofthe field propagating inside the oscillator cavity [5]. Never-theless, an adequate description of this regime can be quiteaccurate in the framework of comparatively simple (1 +1)-dimensional models taking into account spatial effects only inthe form of some effective self-amplitude modulation (SAM)in time domain (see, for example [17–19]). Usually, SAM isassumed to be proportional to the instant power P(t) (t is thelocal time in a frame of reference moving with the group vel-ocity of the pulse). Then the pulse formation and propagationcan be described in the framework of so-called cubic nonlin-ear CGLE [17].

This equation has a stationary solitary wave solution inthe form [17, 20]: A(t) = sech(t/τ) exp(iφ), where A(t) isthe slowly varying field amplitude, τ is the pulse width andφ = ψ ln[sech(t/τ)] is the time-dependent phase with thekey parameter ψ (chirp parameter). A stationary pulse existsonly under condition of a mutual compensation of the phasechanges originated from GDD and SPM. This requires β < 0(β is the GDD coefficient, see below), i.e., anomalous net-dispersion, for the chirp-free (ψ = 0) propagation (the phasebalance equations for the cubic nonlinear CGLE can be found,for example, in [5]). On the contrary, in PDR (β > 0) the phase

balance can be achieved if and only if ψ2 > 2. Thus, a station-ary pulse in PDR has to be chirped [17].

However, numerical simulations have demonstrated thatthe chirped pulse loses its stability at high energy [14]. It isobvious, that the cubic approximation for SAM is flawed inthe high pulse energy (> 100 nJ) range, where the pulse peakpower P(0) approaches Pcr. For KLM this means that over-lap between the pump and laser beams becomes worse [21],i.e., SAM is suppressed (in other words, saturated). A sim-ple generalization taking into account higher-order nonlin-ear effects in this case is to introduce the term ∝ P(t)2,which provides the SAM decrease for higher pulse powers.A similar generalization can also be appropriate for descrip-tion of other mode-locking techniques (additive-pulse mode-locking, mode-locking produced by the intensity-dependentchanges in the field polarization, etc.).

As a result, we come to the cubic-quintic nonlinearCGLE [16] (basic abbreviations and symbols used in the art-icle are summarized in Table 1):

d A

dz=

(−σ + (α+ iβ)

d2

dt2+ (κ − iγ)P −κζP2

)A , (1)

where z is the longitudinal coordinate normalized to the dou-ble cavity length Lcav, α is the square of the inverse spectralfilter bandwidth (as a matter of fact, it is the square of theinverse gain bandwidth for the most of oscillators with broad-band solid-state active medium), β is the GDD coefficient,κ is the SAM coefficient describing enhancement of the net-gain for high-power signal in comparison with low-powerone (cubic on the field SAM), γ is the SPM coefficient ofthe nonlinear element (crystal for a solid-state oscillator, orfiber for a fiber laser). The SPM coefficient in the case, whenthe oscillator mode area S(z) varies inside nonlinear medium,can be defined as (2πn0n2/λ0)P(t)

∫ dzS(z) ≡ γP (t), with in-

tegration over the nonlinear medium length, n0 and n2 arethe linear and nonlinear refractive indexes, respectively; λ0

is the wavelength corresponding to the maximum gain. TheSPM coefficient has maximum γmax = (2π/λ0)

2n2n0 if theconfocal length inside the crystal is much less than the crys-tal length. Growing both, the confocal length and the modearea decrease parameter γ . Parameter ζ describes saturationof SAM in the high-power limit (quintic on the field am-plitude SAM). The SAM parameters can be estimated onlyindirectly from the comparison of the calculated pulse param-eters (spectral width, chirp, etc.) with those obtained from theexperiment [14].

Parameter σ(E) = l − g(E) describes spectrally indepen-dent net-loss in the oscillator (including output loss). Herel is the linear (power- or energy-independent) loss, g(E)

is the gain saturated by the overall intracavity energy E =∫ Tcav/2−Tcav/2 P(t ′)dt ′ (Tcav is the cavity period).

Equation (1) is nonintegrable in the general case and onlyits partial stationary (i.e., z-independent) solutions with par-ticular time-dependent phase profile are known [16]. We shallnot impose constraints on the phase profile and consider thegeneral stationary solution of (1) in the form:

A(t, z) = √P(t) exp (iφ(t)− iqz) , (2)

KALASHNIKOV et al. Chirped-pulse oscillators 505

where q is the phase due to slip of the carrier phase with re-spect to the envelope [5]. We shall consider approximate butquite accurate solution of (1) in the limit of |ψ| 1 [15],which describes CSP. From the balance equations followingfrom the low nonlinear limit of (1) (see [5]), this requiresα/β 1 and κ/γ 1, i.e., there is a strong domination ofGDD over spectral filtering as well as SPM over SAM. Sucha requirement is valid for all known oscillators producing CSPin PDR (see, for example [14]).

Substitution of (2) in (1) gives:

γP(t) = q −βΩ(t)2 ,

β

(dΩ(t)

dt+ Ω(t)

P(t)

d P(t)

dt

)= κP(t) [1 − ζP(t)] −σ −αΩ(t)2 ,

(3)

where Ω(t) ≡ dφ(t)/dt is the instant frequency and the small-ness of the terms ∝ α/β and ∝ d2√P(t)/dt2 has been takeninto account (see [15]). Since P(t) > 0 and dΩ/dt < ∞ (reg-ularity condition), we have from (3): Ω(t)2 < ∆2 ≡ q/β and

γP(0) = β∆2 = 3γ

4ζ(1 − c/2 ∓

√(1 − c/2)2 −4σ(E)ζ/κ) .

(4)

Here ∆ is the parameter of the spectral truncation so thatvalues ±∆ define the borders of the spectrum measured fromits center. Out of these borders the spectral components arelacking. Truncation of the spectrum is the most characteris-tic feature of CSP (see below). Parameter c = αγ/βκ definesa contribution of spectral dissipation (dispersion) relative tothat of SAM (SPM).

The regularity condition dΩ/dt < ∞ allows simplify-ing (3):

dΩ

dt= βζκ

3γ 2

(∆2 −Ω2) (

Ω2 +Ω2L

),

βΩ2L = γ

ζ(1 + c)− 5

3γP(0) . (5)

As a result of integration, there is the implicit expression forthe power time-profile of CSP:

γP(t) = q −βΩ2(t) ,

t = τ

[arctanh

(Ω

∆

)+ ∆

ΩLarctan

(Ω

ΩL

)], (6)

where q = γP(0), Ω(t) ≡ dφ(t)/dt is the instant frequency,τ = 3γ 2/

(ζβκ∆(∆2 +Ω2

L)).

As the phase φ(t) is a rapidly varying function in the limitof |ψ| 1, the expression for the spectral amplitude E(ω) =∫

dt√

P(t) exp (iφ(t)) exp (−iωt) allows obtaining the spec-tral power by the method of stationary phase [15, 22]:

p(ω) ≡ |E(ω)|2 6πγ

ζκ

θ(∆2 −ω2)

ω2 +Ω2L

, (7)

where θ(x) is the Heaviside function.From (7) one can see that the spectral profile of CSP is the

Lorentzian function (with half-width ΩL) truncated at ±∆.

Equation (7) gives the pulse energy:

E =∞∫

−∞dtP(t) =

∆∫−∆

dω

2πp(ω) = 6γ

ζκΩLarctan

(∆

ΩL

). (8)

And, finally, we give an expression for such a physicallyimportant parameter, as the spectral chirp Q(ω):

Q(ω) = (1/2)d2ϕ(ω)

dω2 3γ 2

2βκζ

1

(∆2 −ω2)(Ω2L +ω2)

, (9)

where ϕ(ω) = −φ(t∗(ω))+ωt∗(ω) is the phase in spectralrepresentation and t∗(ω) is defined as the right-hand side ofsecond (6) with replacement of Ω by ω.

The gain dependence on energy is defined by gain mediumkinetics, oscillator mode configuration inside the activemedium, etc. A stable pulse can exist if the saturable net-lossσ(E) is less than the SAM value. The saturable SAM reachesits maximum value κ/4ζ at the peak power P(0)max = 1/2ζ .Since this power cannot excess the critical power of self-focusing Pcr λ2

0/2πn2n0 2π/γmax 1.3 MW, there ex-ists a rigid limitation on the maximum SAM value, whichis less than κP(0)max/2 < κPcr/2 < πκ/γ 1. As a result,σ(E) < πκ/γ 1. This allows expanding σ(E) in series onE around σ(E∗) ≡ 0, where E∗ ≡ PavTcav is the energy storedinside the oscillator cavity in cw regime; Pav is the intracav-ity cw power. In the case of a solid-state oscillator, when theactive medium is much shorter than the beam confocal lengthinside the crystal, E∗ = Is STcav

( g(0)

l −1)

(Is is the gain satura-tion intensity, g(0) is the gain for a small signal) [18].

Thus we can write:

σ(E) ≈ (E −E∗) δ , (10)

where δ ≡ dσdE

∣∣E=E∗ is the “stiffness coefficient” defining the

response of the saturated active medium in the vicinity ofσ(E∗) ≡ 0. For the relation between the confocal and crys-tal lengths considered above, we obtain δ = g(E)2/g(0)E∗ =l2/g(0)E∗.

Equations (10), (4)–(9) give a full analytical represen-tation for the time and spectral profiles as well as the pa-rameters of CSP. These equations show that CSP is com-pletely defined by the dimensionless parameters: P′ = Pζ ,∆′ = ∆

√αζ/κ, Ω′

L = ΩL√

αζ/κ, τ ′ = τ(κ/γ)√

κ/αζ , Q′ =Qκ2/αγζ and E ′ = E(κ/γ)

√ζκ/α. Although master equa-

tions (1), (10) include seven parameters, the CSP parametersdepend on only two dimensionless values: c = αγ/βκ and b ≡σζ/κ = a(E ′/E∗′ − 1), where the dimensionless “stiffness”parameter a ≡ E∗(ζ/κ)δ = l2ζ/g(0)κ (the last equality is validin the case considered in [18]) and E∗′ = E∗ (κ/γ)

√ζκ/α.

Since b depends on the CSP energy, it is necessary to solvetranscendental equations (4), (5), (8) in order to find the pulseparameters. This requires also knowing the gain saturationlaw, i.e. the value of a. However, as it was found, the a value al-most does not affect the CSP parameters. Hence, the presentedanalytical solution for CSP can be evaluated through the twoparameters c and E∗′

.Figure 1 shows the master diagram of borders of the pulse

stability (b = 0, solid curve), between “+”-and “−”-branches

506 Applied Physics B – Lasers and Optics

FIGURE 1 Master diagram for CSP stability border (solid curve), divisionbetween “+”- and “−”-branches of (4) as well as finger- (f) and parabolic-like (p) spectra. Asterisks, squares and points correspond to the numericalresults for the stability border adapted from Fig. 2a, b and c, respectively.Points (A) and (B) correspond to parameters of spectra in Fig. 3a and b, re-spectively. Gray curve shows the points of maximum σ for given E∗′

. Opencircles show two contours of constant b. Parameter a = 0.7

of (4) ((1 − c/2)2 −4b = 0, dashed curve), finger- (∆ > ΩL)and parabolic-like (∆ < ΩL) spectra (b = 3(1 − c2)/16, dot-ted curve) on the dimensionless plane (c–E∗′

). Gray curveshows the points of maximum σ for given E∗′

that is the curveof maximum stability against cw generation (see next sec-tion). The master diagram provides the full description of theCSP properties for all possible choices of parameters. Thedimensional representations of the master diagram will beconsidered in the next section.

3 CSP parameters and stability

As it was mentioned above, the stability of CSP isthe main challenge for the regime under consideration. Thereare two main sources of the pulse destabilization for all mode-locking techniques: i) background radiation growth causingcw-generation or multipulsing [23] and ii) pulse collapsecausing chaotical pulsations [24]. The main criterion provid-ing the pulse stability against the first destabilizing source isσ > 0. The issue concerning the second destabilizing sourcecan be only settled on the basis of numerical simulations.

In order to examine the CSP stability in PDR and the va-lidity of the approximations providing an analytical approach,we have searched the solitary wave solutions of (1) numer-ically. The numerical technique has been based on the split-step Fourier method with the mesh of 1 fs×217 in the time do-main and the maximal longitudinal spatial step of 10−3Lcav

providing a convergence of the numerical procedure for a caseof |ψ| 1.

It has been found, that the saturation of SAM leading tothe limitation of the maximum intracavity P(0), provides theCSP stability against pulse collapse and chaotical pulsation.Simultaneously, CSP is stable only if δ > 0, i.e., the gain sat-uration is required. If ζ = 0 and δ > 0, the remaining sourceof the CSP instability is cw amplification, which takes placewhen σ < 0.

FIGURE 2 Stability borders (solid curves) corresponding to σ = 0, bordersbetween “+” and “−”-branches of (4) (dashed curves) and borders betweenfinger-(f) and parabolic-like (p) spectra (dotted curves). Gray solid curvescorrespond to the parameters providing the maximum CSP energy for a givenintracavity E∗. Crosses show the contours of constant σ . δ = 0.455 µJ−1,γ = 4.55 MW−1, α = 1.1 fs2. (a): κ = 0.04γ , ζ = 0.6γ ; (b): κ = 0.04γ ,ζ = 0.1γ ; (c): κ = 0.02γ , ζ = 0.6γ . Square in (a) and triangle in (c) corres-pond to the experimental probes for CSP (the widest and the most narrowspectra in Fig. 7, respectively)

The simulations have revealed very weak dependence ofthe solution on the stiffness coefficient; therefore, we usedfixed values of δ for a presentation of the dimensional results.Solid curves in Fig. 2a–c show the border of the CSP stabilityagainst the cw growth, which is obtained from the analyti-cal estimation of σ = 0 (this is the dimensional representationof solid curve in Fig. 1). The border in figures correspondsto the minimum GDD β or the maximum energy E∗ provid-ing the pulse stabilization. The points trace the stability bor-der obtained from the numerical simulations (we have usedγ = 4.55 MW−1 and α = 1.1 fs2 that corresponds to Ti:Sa os-cillator [14]). There is a perfect coincidence of the stabilityborders obtained from the numerical simulations (points) withthose obtained from the analytical model presented in the pre-vious section. The tendency towards lower stability at lowGDD is the characteristic feature of CSP (see next section as

KALASHNIKOV et al. Chirped-pulse oscillators 507

well as [11, 14]. The parameters of Fig. 2a are close to thosefor the oscillator presented in [10] (see also [14]).

Decrease of ζ-parameter causes widening of i) the CSPstability range (Fig. 2b), ii) its spectrum and iii) shorteningof the pulse (Fig. 4). The ζ-parameter decrease correspondsto lower SAM saturation, i.e., to operation with P(0) close toPcr. In the case of the focal plane placed inside the crystal,the maximum P(0) = 1/ζ can be only less than Pcr (otherwisethe beam will collapse). Since γ < γmax = 2π/Pcr, the estima-tion for the minimum ζ/γ is 1/2π ≈ 0.16. If the focal planeis located out of the crystal (i.e., the beam inside the crystalis wider), ζ-parameter decreases but simultaneously with γ .Hence, ζ/γ does not decrease essentially in this case, as well.

Decrease of the SAM parameter κ leads to lower stabilityof CSP (the stability border shifts to larger β in Fig. 2c). In thiscase the value of σ is low in a wide range of parameters andconvergence to stationary solution is slow.

Dashed curves in figures mark the borders between the“+” and “−”-branches of the analytical solution (4). Only“−”-branch has an asymptote for ζ → 0 corresponding to thesolitary wave solution of the cubic nonlinear CGLE [18, 20].Crosses in figures show two contours of constant σ . One cansee, that the positive branch provides energy scalability forconstant GDD (almost horizontal branch of σ ∝ (E −E∗) =const). Simultaneously, the negative branch corresponds toa weak dependence of the CSP energy on GDD (almost verti-cal branch of σ = const).

As mentioned in previous section, the spectrum of CSPcan be represented by the Lorentzian function with half-width ΩL, which is symmetrically truncated at frequency±∆. Figure 3 shows spectral profiles for two distinct situa-tions: a) ΩL > ∆, parabolic-like (p) spectrum and b) ΩL < ∆,finger-like (f) spectrum. Comparison of the analytical pro-files (7) with the ones obtained numerically (solid curves andgray circles in Fig. 3, respectively) demonstrates good agree-

FIGURE 3 Parabolic-like (a) and finger-like spectra (b) from (7) (solidcurves) and from the numerical simulation (circles). (a): E∗ = 180 nJ,β = 55 fs2; (b): E∗ = 534 nJ, β = 60 fs2. Other parameters as in Fig. 2a (alsosee points (A) and (B) in Fig. 1)

ment between them. This validates the analytical solution rep-resenting CSP.

Borders between the parabolic-like (p) and finger-like (f)spectra are shown by dotted curves in Figs. 1 and 2. Energygrowth and the approach of GDD to the stability border forsufficiently large E∗ both lead to the finger-like spectrum.

Gray solid curves in Figs. 1 and 2 show the GDD valueproviding the maximum intracavity CSP energy E at fixed E∗.As a result of (10), this curve gives GDD providing the max-imum stability against cw. Figures show that the maximumstability belongs to the “−”-branch of solution (4) and to theparabolic-like spectra for the lower energies and to the finger-like ones for the higher energies. Pulse energy grows almostlinearly with E∗ along this curve.

In agreement with the experiment [11], the spectral widthdecreases and the pulse width increases with GDD (seeFig. 4). The analytical data for the spectral width 2∆ and du-ration T ≈ 1.76τ (points) coincide with the numerical ones(solid curves). The spectral width in the numerical simula-tions has been defined as the width at 1/10 level from thepeak spectral power. For smaller GDDs the analytical esti-mation for ∆ is slightly lower than that from the numericalsimulations. This is a manifestation of the fact that the spec-tral truncation is not so sharp as it follows from the method ofstationary phase ((7), also see Fig. 3). However, this disagree-ment is practically negligible.

Equation (9) demonstrates that: i) parabolic-like spectrumhas the chirp minimum at the central wavelength, ii) finger-like spectrum has the local chirp maximum at the central

FIGURE 4 Numerical dependencies of the spectral width 2∆ and the CSPwidth T on GDD for E∗ = 440 nJ. The points correspond to the spectralwidths and T obtained from (4), (6), (8), (10). Values of κ and ζ are markedabove the pictures. δ = 0.455 µJ−1

508 Applied Physics B – Lasers and Optics

wavelength, iii) CSP with ∆ = ΩL (that corresponds to theparameters of the dotted curves in Figs. 1 and 2) possessesthe most smooth dependence of Q on the frequency in thevicinity of the central wavelength. Such smooth dependence ismost appropriate for further compression of CSP aimed to ob-tain femtosecond chirp-free pulse [11, 14]. The compressionfactor from the picosecond chirped to the femtosecond chirp-free pulse is 3γ/κ so that the sub-30 fs pulses are achiev-able. As a consequence of such compression, P(0) increases 3γ/κ-folds and this provides over-10 MW peak powers di-rectly from an oscillator.

Some additional comments are required for comparisonof the considered regime with that formed by the self-similarpulses [13]. The SAM parameters of Fig. 2a, c and Fig. 4acorrespond to comparatively narrow spectra and long pulses(Fig. 4a). In this case β∆2 1 and the pulse chirp variationalong the cavity is small. Hence, the model with uniformlydistributed GDD (1) is valid. However, the spectra widen andthe pulses shorten with the ζ/γ decrease (Fig. 4b, see also (3)).The product β∆2 becomes 1 and the pulse parameters vari-ation along the cavity must be taking into account. Figure 5shows comparison of the cases with different GDD distribu-tions. Solid curves correspond to simulations based on (1).Dashed curves are obtained with taking into account the realpositive dispersion of a 3 mm Ti:Sa crystal and the negativeGDD, which is symmetrically distributed around the crys-tal. Due to partial chirp compensation caused by the negativeGDD, the pulse shortens. The maximum variations of pulsewidth along the cavity are 5%–35%. As the reduced T in-creases the spectral width, the stability is lowered by the spec-tral loss. When all negative-GDD is placed at only one sideof the crystal, the pulse becomes even shorter (dotted curve),however, it loses its stability for higher energies (the regime islacking for the parameters corresponding to curves 2).

Hence, the concept of self-similar pulses begins to work,when the SAM saturation is reduced or SPM increases, i.e.,when ζ/γ -ratio is small. For instance, the SPM parameter γ

grows essentially in fiber lasers, where the nonlinear propa-gation distance is large (meters). As a result, the self-similarpulses develop [12].

FIGURE 5 Numerical dependencies of the intracavity pulse width T onnet-GDD for κ = 0.04γ , ζ = 0.16γ . E∗ = 430 nJ (gray curves 1), 860 nJ(black curves 2). The meaning of the curves is explained in the text

In the next section we shall demonstrate that the parame-ters of our Ti:Sa chirped-pulse oscillators allow the descrip-tion based on the concept of CSP (i.e., on the distributedmodel (1)).

4 Experimental observations of CSP and discussion

10 MHz Ti:Sa oscillator has already been de-scribed in [10, 11]. It has a long cavity (≈ 15 m), beingcompact at the same time due to incorporated delay line.Accurate dispersion adjustment is provided by chirped mir-rors and intracavity prisms. The oscillator generates spectrum≈ 90 nm and 26 fs pulses (after extracavity compression).Output power (pulse energy) is almost proportional to thepump, at least at moderate pumps from 4 to 8 W. The max-imum output pulse energy in our case was ≈ 150 nJ. At thevery first attempts to obtain the highest pulse energy it wasfound that one has to adjust properly the cavity dispersion asthe pump and pulse energy grows. Without such adjustmentthe oscillator falls into a mode specific for the negative dis-persion regime, with smooth broad spectrum. This regime isunstable, showing strong deviations of the pulse-to-pulse en-ergy. By increasing the amount of intracavity glass, one canreturn to the characteristic sharp-edged spectrum, which cor-responds to excellent pulse-to-pulse stability.

Dependence of the necessary amount of positive net-GDD(to keep pulses stable) for different pumps (and, correspond-ingly, CSP energy) is shown in Fig. 6. Bars show the relativeaccuracy of the GDD measurements. One can see, that GDDvalue required for the pulse stabilization against cw radiation,grows with the energy. This agrees with the main intentionof the theoretical analysis of the CSP stability: GDDs corres-ponding to the stability threshold and to the maximum pulsestability increase with energy (solid black and grey curves inFig. 2, respectively). Worth noting that not only the optimumGDD increases with the pump energy in the experiment, butalso the range of dispersions, corresponding to stable regimeof mode-locking, becomes narrower.

The experimental spectra are shown in Fig. 7. These spec-tra belong to the experimental curve in Fig. 6. As it follows

FIGURE 6 The measured dispersions providing stable CSPs in dependenceon the output pulse energy

KALASHNIKOV et al. Chirped-pulse oscillators 509

FIGURE 7 Measured spectra corresponding to the data of Fig. 6. Intracav-ity energies corresponding to the spectra are 180 (solid), 300 (dashed) and360 (dotted) nJ, respectively

from (5), the spectra have the truncated edges (see also Fig. 3).The theoretical spectrum Fig. 3a is close to the widest spec-trum in Fig. 7 (parameters correspond to point A in Fig. 1 andto square in Fig. 2a, respectively). The experimental spectralshapes can be attributed to the parabolic-like type.

Equations (3), (5) and (7) allow estimations of the CSP pa-rameters corresponding to the spectra from Fig. 6. Cardinalproblem is that the SAM parameters are not directly mea-surable. For example, their values corresponding to Fig. 2aprovide an excellent agreement with the width of the widestspectrum shown in Fig. 7 by solid curve (see square in Fig. 2,which lies precisely on the curve of maximum stability). Inthis case the GDD value is comparatively small and the spec-tral edges are enhanced (see solid curve in Fig. 6). This phe-nomenon can be explained by action of the fourth-order dis-persion [14], which is not taken into account in (1).

In agreement with the theoretical data (Fig. 4), the spec-trum becomes narrower as GDD grows. However, the spec-trum shortening with GDD is more rapid in experiment thanit follows from the model with fixed κ and ζ . This can beexplained by variation of the SAM parameters in experi-ment. The reason is that one has to change slightly the activemedium position with the pump change in order to obtain theself-starting KLM. If the κ-parameter decreases and the ζ-parameter increases, spectrum becomes narrower for fixed β

and E∗. For example, the spectral width and the energy of themost narrow spectrum in Fig. 7 as well as the correspondingGDD can be obtained theoretically within the 20% accuracyfor κ = 0.02γ and ζ = 0.6γ (triangle in Fig. 2c). Hence, thesystem operates in the vicinity of maximum stability curve inthis case, too. Theoretical estimations suggest that it is alsoright for another experimental points of Fig. 6.

Analysis of the highest pulse energy reachable for CSPneeds taking into account the dynamical gain saturation, i.e.,the dependence of σ(E) on the time, as there is a time-dependence of E(t) = ∫ t

−∞ P(t ′)dt ′ in general case. Sucha time-dependence causes the gain variation from the pulsefront to its tail. For stationary pulse propagation this variationcan be estimated as ≈ g(0)Tcav/2Tr, where Tr is the gain re-

laxation time. Surplus gain can result in noise amplification infront of the pulse. Therefore we have the exacting criterion forthe pulse stability:

σ (E) >g (0) Tcav

2Tr. (11)

One can see, that the minimum σ providing pulse stabilizationincreases with pump (i.e., g(0)). The contours of constant σ

in Fig. 2 demonstrate, that the stability borders shift to higherenergies as σ-parameter increases. These contours cut off therange of lower E∗ and 1/c (i.e., β), which cannot already pro-vide the stable pulses and, thereby, the stability range reduceswith the pump growth.

Since the output energy E out ≡ Poutav Tcav (Pout

av is the outputaverage power), we have from (11):

E out <2Pout

av Trσ (E)

g (0). (12)

After extra-cavity chirp compensation, the compressedpulse has width T > 2/∆. CSP with the parabolic-like spec-trum has a chirp with comparatively smooth dependence onfrequency, and therefore it is better compressible. For theparabolic-like spectra we have ∆ < ΩL and arctan

(∆ΩL

)<

π/4 in (8). This and (8) give:

E <3πγ

4ζκT . (13)

Upper bound of this inequality is reached at the f –p bound-ary, i.e., when ∆ = ΩL.

Since σ cannot exceed the maximum SAM, σ < κ/4ζ .Therefore, (12), (13) as well as E out ≡ Eχ (χ is the outputmirror transmission coefficient) give:

E out <

√3πγχ

8ζ2g (0)Pout

av T Tr . (14)

For the finger-like spectra T > 2/ΩL, the maximum σ is(9/16)κ/4ζ and arctan

(∆ΩL

)< π/2. Then (14) is valid for this

type of CSP, too.As γ < 2π/Pcr and 1/ζ < Pcr, our estimation for the max-

imum output pulse energy is:

E out <

√3π2χ

4g (0)Pout

av PcrT Tr √10Pout

av T [J] . (15)

In the last expression we used the Pcr = 1.3 MW and Tr =3.5 µs values for Ti:Sa and χ/g(0) ≈ 1/4. For our oscillatorswith Pout

av = 2 W (1 W) and T ≈ 40 fs, we have E out < 0.9 µJ(0.7 µJ) in good agreement with the maximum energy levelreported in [11] (≈ 500 nJ, T ≈ 45 fs, the pulse repetitionrate is 2 MHz). As it follows from theoretical estimations,the maximum energy and compressibility are reachable inthe vicinity of the curve of maximum stability and of theborder between p- and f -spectra (see Fig. 1) for parametersζ → ζmin = 1/Pcr and γ → γmax = 2π/Pcr.

Simultaneously, since σ > gE out/2Poutav Tr 0.1 for g = 1

and Poutav = 1 W from (12) [11, 14], one has to use saturable

510 Applied Physics B – Lasers and Optics

Bragg reflector in order to provide such a high value of SAM.In this case the optimum pulse repetition rate of an oscilla-tor is 1/Tcav = Pout

av /E out 2.2 MHz (1.4 MHz) that is close to2 MHz of [11].

5 Conclusions

We presented the theory of a chirped-pulse oscilla-tor. The pulse formed in PDR can be described as a solitary-wave solution of the nonlinear cubic-quintic CGLE. We pro-posed a technique allowing one to find analytically the ap-proximate closed-form solution corresponding to CSP, so thatthe oscillator can be characterized by only two dimensionlessparameters combining GDD, gain bandwidth, SPM and SAMcoefficients as well as cw energy and “stiffness” parameter ofthe gain saturation.

It was demonstrated analytically that the CSP spectralprofile is Lorentzian with the truncation at some defined fre-quency ∆ distant from the carrier frequency. Ratio of theLorentzial spectral half-width to ∆ divides all possible spec-tral shapes into two main classes: parabolic-like and finger-like spectra. The last class corresponds to lower GDDs andhigher energies.

Numerical simulations demonstrated an excellent agree-ment with the results of analytical treatment of CSP. Theobtained analytical solutions were found to be stable withina wide range of parameters. The CSP destabilization dueto cw-growth has been investigated. Variations of the CSPparameters (spectrum broadening and pulse shortening withpositive GDD approaching zero) were described analyticallyand confirmed numerically, as well.

We presented also the experimental observations of ≤150 nJ pulses from the 10 MHz Ti:Sa oscillator operating inPDR. Spectral shapes, widths and pulse durations are closeto those obtained theoretically. It was found, that the os-cillator operates in the vicinity of the maximum stabilitycurve calculated analytically and corresponding to the max-imum saturable net-loss coefficient σ for a given pump power.The dispersion providing the stable CSP grows with energywhereas the dispersion range providing CSP stability be-comes narrower.

The solution for CSP obtained analytically allowed es-timation for the maximum pulse energy reachable in PDR.Factors confining the pulse energy are: i) approaching thepeak power to the critical power of self-focusing; ii) cw-amplification at the pulse front. The last factor confines themaximum cavity period and, thereby, the pulse energy forthe fixed Pav and SAM. As the growth of Pav is limited for

a given oscillator geometry and pump, the further pulse en-ergy growth is possible only due to oscillator lengthening.However, this needs the SAM strengthening to suppress cw-amplification. This is possible only with semiconductor Braggreflector as mode-locker.

ACKNOWLEDGEMENTS The authors acknowledge the helpof A. Fernandez in experimental part of the work and the support of theChristian Doppler Society and the Max-Planck Gesellschaft. Author (V.L.K.)acknowledges the support from the Austrian National Science Fund (FWFproject P17973).

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