Energy scalability of mode-locked oscillators: a completely analytical approach to analysis

V. L. Kalashnikov; A. Apolonski

doi:10.1364/OE.18.025757

1. Introduction

Development of 1 to 100-μJ-level pico- (ps) and femtosecond (fs) modelocked oscillators can provide a good alternative to MOPA (master oscillator-amplifier) or CPA (chirped-pulse amplification) [1] systems usually operating at kHz repetition rates: the oscillator is intrinsically compact, stable, simple and less expensive. MHz repetition rates of such oscillators will lead to substantial reduction of the measurement time in pump-probe experiments as well as to a better signal-to-noise ratio in nonlinear experiments [2].

Existing energy-scalable oscillators can be divided into three main types: i) thin-disk mode-locked oscillators operating in both anomalous (ADR; the main abbreviations and symbols are defined in Table 1) [2–4] and normal (NDR) [5] dispersion regimes; ii) all-normal-dispersion (ANDi) fiber oscillators [6, 7]; and iii) solid-state chirped-pulse oscillators (CPOs) operating in the NDR [8, 9]. To date, over-10-μJ fs-pulses have been obtained directly from Yb:YAG thin-disk oscillators [3, 4], sub-1-μJ pulses from Ti:Sa CPOs [8], and over-100-nJ pulses from the ANDi fiber oscillator [7]. High-energy fs-pulses nowadays allow direct experiments on light-matter interactions at intensity levels approaching PW/cm² [10, 11]. In particular, high-harmonic generation at such energy levels promises development of table-top VUV/XUV sources, which are of interest for physics, chemistry, material science, medicine, and biology.

Table 1. Main abbreviations and symbols

View Table | View all tables in this article

Existing theories of fs-pulse energy scalability are based on the dissipative soliton (DS) concept [12] of the complex nonlinear Ginzburg-Landau equation (CNGLE) [13–15, 17, 43]. As was found [14], the energy scalability of the DS is substantially enhanced in the NDR due to strong pulse chirp, which results in DS stretching. Such stretching reduces the DS peak power, thereby causing pulse stabilization. Simultaneously, a strong chirp provides a sufficiently broad spectrum allowing pulse compression by a factor of 10–100.

Approximated integration of the CNGLE supported by extensive numerical simulations have allowed the problem of the energy-scalable DS to be presented in the form of two-dimensional master diagrams connecting the soliton energy with the universal laser parameter c ≡ τΣγ/|β|ζ, where τ is the squared inverse bandwidth of the spectral filter, Σ is the net-loss coefficient, γ is the self-phase modulation (SPM) coefficient, β is the net-group-delay dispersion (GDD) coefficient (the NDR, i.e. β < 0 was assumed for the approach under consideration), ζ is the self-amplitude modulation (SAM) coefficient [15]. As was pointed out, the DS is perfectly energy-scalable if there is a constant nonzero limit of c for E → ∞ (E is the DS energy) [18, 19]. Physically this means that there is no need for the variation of the laser parameters constituting the c-parameter in order to provide the DS energy scaling. It was also found, that such perfect scalability is possible for the cubic-quintic CNGLE [19], which describes, for instance, a Kerrlens mode-locked solid-state oscillator, but it is not possible in the case of the generalized CNGLE describing an oscillator mode-locked by a semiconductor saturable mirror (SESAM) [18, 20]. On the other hand, the latter type of oscillators is of most interest for applications.

The energy-scalable DS concept is closely coupled with the so-called DS resonance concept, which is based on numerical integration of the cubic-quintic CNGLE supported by the approximated solution by means of the method of moments [17, 21]. The advantage of this approach is that it can also be applied to the ADR (i.e. β > 0), where the energy-scalable DS exists as well [22]. Nevertheless, there are some disadvantages with the approach of [17,21]: the space of DS resonances is multi-parametrical and is not directly connected with existing energy-scalable oscillators.

In this article we present a completely analytical approach to analysis of the energy-scalable DS. The approach is based on the variational model of the DS of the generalized CNGLE and is validated by numerical simulations of thin-disk Yb:YAG oscillators operating in both AND and NDR. In spite of the model in [20], that presented here is not limited by the conditions β < 0 (i.e. only NDR), τ Σ ≪ |β| (domination of dispersion over spectral dissipation), and γ ≫ ζ (domination of SPM over SAM). As a result, all types of energy-scalable oscillators can be analyzed from a unified viewpoint. Simultaneously, the theory presented here circumscribes the validity of the model of [20] in the NDR. The most interesting properties of the model presented are its reduced parametrical space and scaling rules. As a result of these properties, an energy-scalable oscillator can be described on the basis of a two-dimensional master diagram connecting the dimensionless DS energy with the universal laser parameter c. Table 1 contains the main parameters and acronyms used in the text.

2. Variational DS solutions of the generalized CNGLE

The variational approach is well-established for soliton-like systems covering optics, condensed-matter physics, etc. (for a review see, for instance, [23]). In optics, the basic idea is that one can describe field-envelope evolution by the force-driven Euler-Lagrange equations (the so-called Kantorovitch method) [23–28]. For the 1+1 dimensional evolution problem (z is the propagation distance, i.e. the cavity round-trip number for an oscillator; t is the local time), the corresponding system of equations is

\frac{\partial \int_{- \infty}^{\infty} Ldt}{\partial f} - \frac{d}{d z} \frac{\partial \int_{- \infty}^{\infty} Ldt}{\partial f} = 2 ℜ \int_{- \infty}^{\infty} Q \frac{\partial a}{\partial f} dt,

where the Lagrangian L corresponds to the conservative factors defining the nonlinear Schrödinger equation (below γ is the SPM coefficient, β is the net-GDD coefficient, so that β < 0 corresponds to the NDR):

L = \frac{1}{2} {i [a^{*} \frac{\partial a}{\partial t} - a \frac{\partial a^{*}}{\partial t}] + [β \frac{\partial a}{\partial t} \frac{\partial a^{*}}{\partial t} - γ | a |^{4}]} .

Here a(z,t) is the slowly-varying field envelope, so that |a|² is the power. The trick is that a(z,t) is assumed to be some trial function approximating a desired soliton-like solution. We chose the following form for this function [29]:

a (z, t) = A (z) exp [i ϕ (z)] sech {[\frac{t}{T (z)}]}^{1 + i ψ (z)},

where the values dependent on the propagation distance are: A is the amplitude, T is the pulse width, ψ is the chirp, φ is the phase shift due to a slip of the carrier phase with respect to the envelope. Then, the arguments of the functional derivatives in Eq. (1) are f ≡ {A, T, ψ, ϕ}.

The next step is to define the dissipative forces affecting the field evolution [30, 31]:

Q \equiv i [- \sum + \frac{ρ}{1 + σ \int_{- \infty}^{\infty} {| a |}^{2} d t} (1 + π \frac{\partial^{2}}{\partial t^{2}}) + \frac{μ ζ {| a |}^{2}}{1 + ζ {| a |}^{2}}] a .

Such an expression corresponds to an oscillator with the net-loss coefficient Σ (including the losses due to the output coupler and the saturable losses), and the low-signal gain coefficient ρ saturable by a total field energy (σ is the inverse gain saturation energy). The spectral dissipation is defined by the parabolic-like gain-band profile [32] with the squared inverse gain bandwidth τ. The SAM with a modulation depth μ providing a mode-locking corresponds to a semiconductor saturable mirror (SESAM) which can be modeled as a perfectly saturable absorber if T (>1 ps for the high-energy oscillators under consideration) is much larger than the SESAM relaxation time (≈ 100 fs) (the adiabatic approximation) [33]. The parameter ζ is the inverse saturation power of a SESAM. Hereinafter, two practically important types of the DS will be considered: i) the chirp-free (ψ ≡ 0) DS in the ADR, and ii) the chirped DS (CDS) in the NDR.

Since the oscillator operates slightly below the CW threshold, one can eliminate the gain parameters from the final equation obtained from Eqs. (1–4): $\frac{ρ}{1 + σ \int_{- \infty}^{\infty} {| a |}^{2} dt} \approx Σ$ . Then, the parameters of the chirp-free DS are

\frac{\partial ϕ}{\partial z} = - \frac{b}{2} A^{2}, T = \frac{1}{A \sqrt{c}},

where the pulse width is normalized to

\sqrt{Σ τ}

, the power DS peak power A² is normalized to ζ, b ≡ γ/ζ, and c ≡ τΣγ/βζ. The dimensionless DS amplitude A can be found from the equation

μ (2 + \frac{ln (\frac{1 + A^{2} - A \sqrt{1 + A^{2}}}{1 + A^{2} + A \sqrt{1 + A^{2}}})}{A \sqrt{1 + A^{2}}}) + 2 Ξ - \frac{2 c A^{2}}{3} = 0,

where the saturated net-gain parameter

Ξ \equiv \frac{ρ}{1 + 2 A^{2} T} - Σ

.

For the CDS, one has

\begin{matrix} \frac{\partial ϕ}{\partial z} = - \frac{b}{24 A^{2}} \frac{12 b A^{4} + ψ (12 A^{2} Ξ + μ π^{2} - 32 c A^{4})}{b - 2 c ψ} . \\ T = 2 A \sqrt{\frac{(1 + ψ^{2}) (2 c ψ - b)}{c ((12 A^{2} Ξ + μ π^{2}) ψ - 4 b A^{4})},} \\ ψ = - \frac{4 b A (A \sqrt{1 + A^{2}} (c A^{2} - 3 (μ + Ξ)) + 3 μ arctanh [\frac{A}{\sqrt{1 + A^{2}}}])}{c (\sqrt{1 + A^{2}} (12 A^{2} (2 μ - Ξ) - μ π^{2})) - 24 A arctanh [\frac{A}{\sqrt{1 + A^{2}}}]} . \end{matrix}

The equation for the amplitude A containing the dilogarithm functions is too lengthy and is omitted [34].

Thus, the DS belonging to an isogain Ξ = const is completely characterized by the c- (for the chirp-free DS) or the c- and b-parameters (for the CDS). The pulse parameters can easily be found by numerical solution of the sole equation Eq. (6) for the chirp-free DS. The corresponding equation for the CDS is presented in [34].

3. Numerical DS solutions of the generalized CNGLE

To validate the analytical solutions obtained and connect them with the real-world laser configurations, we performed numerical simulations based on the undistributed models of a high-energy oscillator mode-locked by a SESAM.

An undistributed evolution of time-(t)-dependent slowly-varying field envelope A(t) is modeled on the basis of the two maps shown in Fig. 1. Such maps represent two thin-disk Yb:YAG oscillators: an airless one (a) and air-filled one (b). The nonlinear loss operator

\hat{L} [A] = A (t) exp [- κ - \frac{μ}{1 + ζ {| A (t) |}^{2}}]

describes the SESAM response under adiabatic approximation, and κ is the unsaturable net-loss (so that Σ ≡ μ + κ). The conservative operator Ĥ describes the SPM in air (for the air-filled oscillator) with the coefficient γ_air and the compensating GDD β_M introduced by the chirped-mirrors. This operator corresponds to the Lagrangian (2).

Fig. 1 Maps for two thin-disk Yb:YAG oscillators: an airless one (a) and air-filled one (b). Ĥ blocks correspond to the delay lines with the dispersion compensators, Ĝ is the gain head, L̂ is the SESAM.

Download Full Size | PDF

The integro-differential stochastic operator Ĝ means

\begin{matrix} \hat{G} [A] = i \frac{β_{g}}{2} \frac{\partial^{2}}{\partial t^{2}} A (t) + i γ_{g} {| A (t) |}^{2} A (t) + \\ (\frac{ρ Ω_{g}}{1 + N σ \int_{- \infty}^{\infty} {| A (t) |}^{2} d t} \int_{t}^{\infty} exp [- Ω_{g} (t' - t)] A (t') d t') + s (t) . \end{matrix}

It describes a gain medium with the GDD coefficient β_g and the SPM coefficient γ_g. The saturable gain with the gain coefficient ρ for a small signal and the inverse saturation energy σ has the causal Lorentz spectral profile [35] with the width Ω_g (so that

τ = Ω_{g}^{- 2}

). N is the number of transits through an active medium for one transit through a thin-disk gain head (N =2 and 1 for maps (a) and (b) in Fig. 1, respectively). The complex stochastic value s(t) is such that [36]

〈 s (t) s^{*} (t') 〉 = 2 Σ θ \frac{h ν}{δ t} δ (t - t')

describes the quantum noise of an active medium (θ is the enhancement factor due to an incomplete inversion of the active medium, δt is the time step in subdividing the time window representing a(t)).

The propagations of the field A(t) through the maps shown in Figs. 1 (a) and (b) are described by

\begin{matrix} A_{z + 1} (t) = [\hat{H} \otimes \hat{H} \otimes \hat{G} \otimes \hat{H} \otimes \hat{G} \otimes \hat{L} \otimes \hat{G} \otimes \hat{H} \otimes \hat{G} \otimes \hat{H} \otimes \hat{H}] A_{z} (t), \\ A_{z + 1} (t) = [\hat{L} \otimes \hat{H} \otimes \hat{G} \otimes \hat{H} \otimes \hat{L}] A_{z} (t), \end{matrix}

respectively.

In the framework of the model under consideration, the relaxation dynamics of gain can be taken into account, as well. In this case, the gain coefficient g in Eq. 9 becomes round-trip dependent:

ρ_{k + δ k} = P T_{r} ρ_{max} + (ρ_{k} - P T_{r} ρ_{max}) exp [- \frac{Δ t}{T_{r}}],

where δk is the part of the round-trip between successive transits through the gain medium (δk = k/4 and k/2 for maps (a) and (b) in Fig. 1, respectively), Δt is the time interval between successive transits through the gain medium (= T_cav/4 and T_cav/2 for maps (a) and (b) in Fig. 1, respectively), P is the pump rate, ρ_max is the maximum gain coefficient, T_r is the gain relaxation time, and T_cav is the cavity period.

The iterative Eqs. (11,12) were solved on the basis of the symmetrized split-step Fourier method on a mesh with 2¹⁷ or 2¹⁸ points and with minimum time steps of δt =2.5–10 fs. Steady-state solutions were attained after 10,000–50,000 round-trips. The main simulation parameters are presented in Table 2. The b′-type oscillator is geometrically identical to the b-one, but corresponds to an oscillator without output. The energy scaling is provided by scaling of the laser beam size, and the cavity period T_cav (which affects the saturation parameter σ) and/or scaling of the pump rate P.

Table 2. Simulation parameters (per round-trip) for the oscillator maps shown in Fig. 1 (a) and (b). Map b′ is geometrically identical to the b one, but has other parameters.

View Table | View all tables in this article

4. Results and discussion

To compare the results of numerical and analytical modeling, we analyzed the parametrical spaces of stable DSs. The simplest criterion of DS stability in the framework of an analytical approach is the so-called vacuum stability of the generalized CNGLE presented by (2, 4). Formally, this means that Ξ < 0, i.e. the saturated net-gain is negative. Under this condition, noise amplification outside the DS is not possible. Previously, such a criterion was numerically validated for both NDR and ADR [14,37]. Thus, the parametrical space of stable DSs is the master diagram of the isogains Ξ = 0.

4.1. Anomalous dispersion regime (ADR)

The master diagrams are two-dimensional for the ADR (β > 0; see Eqs. (5,6)) and can be plotted in the space of the dimensionless parameters c ≡ τΣγ/βζ versus the energy E normalized to $ζ / \sqrt{Σ τ}$ . The sector of such a master diagram is presented in Fig. 2. It is drawn in the vicinity of the numerical stability threshold (lines+symbols) of the thin-disk Yb:YAG oscillators corresponding to the a, b, and b′-maps in Fig. 1 (see also Table 2). Stable pulses exist below the corresponding threshold curves. This means that the isogain Ξ = 0 corresponds to the maximum c for a given energy E. Physically speaking, there is a minimum GDD for a given energy, gain bandwidth, net-loss, and ratio of the SPM to the SAM.

Fig. 2 Thresholds (c ≡ τΣγ/βζ vs. E) of the pulse stability (solid curves) and the pulse widths T_FWHM at these thresholds (dashed curves) in the ADR. The different colors correspond to μ=0.002 (black), 0.0025 (green), 0.004 (red), 0.005 (magenta), and 0.008 (blue). Analytical thresholds correspond to the lines without symbols, numerical ones correspond to the curves with symbols. The dimensional values of the GDD and the output pulse energies at the extreme points are shown for oscillator types a and b. For an oscillator of the b′-type, the intracavity energies are shown. The pulses are stable below the thresholds shown.

Download Full Size | PDF

One can see from Fig. 2 that the analytical stability thresholds (solid lines without symbols) correspond to a linear dependence of c on E within the region of the master diagram shown. The pulse width remains almost constant. This means that the energy scaling results from a peak-power scaling (the dependence would be quadratical for the scaling of the pulse width [18]). As the pulse width is comparatively small (T_FWHM is normalized to $\sqrt{Σ τ}$ ), the peak power is high and, as a result, the compensating GDD is huge for the oscillator types a and b considered.

The master diagram suggests extremely simple scaling rules. Energy growth requires the following steps for pulse stabilization: i) proportional growth of the GDD; or ii) proportional decrease of the SPM (helium-purified or airless oscillators). Simultaneously, change of the SAM (ζ-parameter) does not affect the stability (by the definitions of c and the dimensionless E) for the other fixed parameters (including the fixed dimensional energy). The gain bandwidth decrease (τ-growth) impairs the stability ( $\propto \sqrt{τ}$ ) and increases the DS width. The growth of the SAM depth (μ-parameter) enhances the stability. However, such a growth can result in Q-switched mode-locking, which is beyond the scope of the analytical model.

The numerical stability thresholds are shown in Fig. 2 by the curves with symbols. As was found, the main destabilization mechanism in the numerical simulations is multiple pulse generation. One can see that the numerical thresholds are in good agreement with the analytical ones, although the numerical model is undistributed and involves a wider class of laser phenomena. There is a slight deviation from the linear scaling law for oscillator type b, which means pulse broadening and, as consequence, stabilization of the DS requires a more rapid growth of the GDD with energy.

It should be noted that the dynamical factor appearing in Eqs. (2,4,8,9) is the intracavity DS power and, as consequence, the intracavity DS energy E. Recalculation of the latter into the output dimensional energy requires, of course, that the transmittance of the output coupler be taken into account. Nevertheless, this transmittance is the dynamical factor, as well, and it contributes through the net-loss parameter Σ. The scaling rules suggest that the Σ-growth impairs the stability ( $\propto \sqrt{Σ}$ ) for the other fixed parameters (including the fixed intracavity dimensional energy). As a result, it is possible to decrease the stabilizing GDD value in oscillator type b′ correlation to oscillator types a, b. Moreover, the pulse width will be shorter for the b′-oscillator because it is proportional to $\sqrt{Σ}$ , and Σ is substantially reduced for an oscillator without output.

Figure 3 demonstrates a full-scale analytical master diagram in the ADR. One can see, that there are two main sectors in the diagram: i) sector of almost constant c (solid lines; c/μ ≈ 1 here) and decreasing T with growing E (dashed curves; left-hand side of the diagram); and ii) sector of linearly decreasing c (solid curves) and almost constant T (dashed curves; right-hand side of the diagram). The DS is stable below the solid curves shown and the stability is enhanced by the modulation depth (μ) growth. The dotted lines show the stability thresholds for the simplest stability criterion c/μ < 1 [38], which is the limiting case (β ≫ τ) of the vacuum stability of the cubic CNGLE [39].

Fig. 3 Thresholds (c ≡ τ Σγ/βζvs. E) of the pulse stability (solid curves) and the pulse widths T_FWHM at these thresholds (dashed curves) in the ADR. The different colors correspond to μ=0.002 (black), 0.008 (blue), and 0.016 (green). Symbols demonstrate the operational points in the vicinity of stability threshold for the different oscillators (corresponding references and output energies are inscribed). Dotted curves correspond to the criterion c/μ=1 [38].

Download Full Size | PDF

Scaling properties differ in these two sectors. The left sector corresponds to small energies (and/or small SAM, i.e. ζ) and/or to strong dissipation (large τ and/or Σ). This sector is perfectly energy-scalable (i.e. c does not depend on E), but at the cost of strong variation of the DS width.

The right sector corresponds to large energies (and/or large SAM, i.e. ζ) and/or to weak dissipation (small τ and/or Σ). In this sector, c decreases linearly with E, but the pulse width T is minimum and remains almost constant.

For clearness, the points corresponding to some oscillators operating in the vicinity of stability threshold are shown by symbols. Unfortunately, the experimental estimations of parameters required for the formulation of the CNGLE were found only for a few oscillators published. Such estimations are afforded if comparison of theory with experiment is available. This fact defines our choice of the data presented in Fig. 3.

The red circle corresponds to oscillator type a (see above). Other points are obtained by recalculation of parameters taken from references (presented in Fig. 3). Since the solitonic fiber oscillator (red star in the figure) involves the use of a spectral filter, it holds that c = τ γ/βζ for it (correspondingly, the normalization does not include Σ for this case). One can see that, despite a huge scatter of the parameters and even the SAM types, the operational points are well-fitted by the master diagram. Also, Fig. 3 demonstrates that the oscillators can operate within both sectors of the diagram, although the right sector is more typical and provides better characteristics (higher energy and lower pulse width).

After all, a pure solitonic fiber laser is inferior to a broadband solid-state one with respect to energy scalability. The linear scaling law in the right-hand sector of Fig. 3 means that γE ∝ β/τ. As a result, the energy growth cannot result from oscillator period scaling only, because γ, E and β would increase proportionally to a fiber oscillator length in this case. Therefore, one should reduce the SPM value regardless of the GDD value, but this is hardly possible for a solitonic fiber oscillator. A possible way of energy scaling is mode-size scaling (i.e. γ-scaling) in a large-mode-area photonic crystal fiber while keeping GDD in the anomalous dispersion range. Another way of getting the energy higher is to reduce the oscillator spectral width (i.e larger τ), which, however, inevitably increases the pulse width. On the contrary, the independent variation of the GDD and SPM can be realized within a substantially broader range of parameters for a broadband solid-state oscillator.

4.2. Normal dispersion regime (NDR)

The sector of the master diagram for the NDR (β < 0) is presented in Fig. 4. It is shown in the vicinity of the numerical stability thresholds (curves + symbols) of the thin-disk Yb:YAG oscillators corresponding to the a and b types (Fig. 1 and Table 2). The analytical thresholds are obtained from the approximated model of Ref. [20]. One can see that there is good agreement of the analytical model with the numerical data and this agreement becomes perfect with the energy growth.

Fig. 4 Thresholds (|c| ≡ τΣγ/|β|ζ vs. E) of the pulse stability. The different colors correspond to μ=0.002 (black), 0.005 (magenta), and 0.008 (blue); b ≡ γ/ζ=2×10⁻⁴ (black and blue), 4×10⁻³ (magenta). Analytical thresholds correspond to the lines without symbols, numerical ones correspond to the curves with symbols. The dimensional values of the GDD and the output pulse energies in the extreme points are shown for oscillator types a and b. The pulses are stable below the thresholds shown.

Download Full Size | PDF

One can see that the GDD level in the NDR is substantially reduced (an approximately tenfold reduction) in relation to that in the ADR. This results from a large pulse chirp causing pulse stretching. The energy scaling is provided mainly by pulse width growth (and, correspondingly, chirp growth). As a result, the dependence of c on E is weaker than that in the ADR. Hence, one can conclude that the NDR is preferable for energy scaling to the ANR.

Figure 5 demonstrates the full-scale analytical master diagrams in the NDR corresponding to the model of Ref. [20] (curves) and the variational model (Eqs. (7); symbols). The model of Ref. [20] assumes domination of nondissipative effects over dissipative ones, i.e. β ≫ τΣ and γ ≫ μζ. The variational approach does not require such an assumption. One can see that the model of Ref. [20] results in overestimation of the DS stability within the region of |c|/μ → 1 when b ≪ 1 (black curve vs. black circles). However, the agreement between the models is quite accurate for |c| ≪ 1 and increases with the b-growth (blue curve and blue triangles).

Fig. 5 Thresholds (|c| ≡ τΣγ/|β|ζ vs. E) of the pulse stability. The different colors correspond to b ≡ γ/ζ=0.001 (black), 0.1 (red), and 10 (blue); μ=0.008. Analytical thresholds from [20] correspond to the curves (the CDS are stable in the direction of the arrows marked in regular type). Analytical thresholds from the variational model correspond to the symbols (the CDS are stable in the direction of the arrows marked in italics).

Download Full Size | PDF

The appearance of the additional degree of freedom (b-parameter) calls for comment. The decrease of ζ (i.e. the b-growth due to suppression of the SAM with simultaneous decrease of the normalized E and the c-growth) almost does not affect the DS stability if other parameters (including the dimensional energy E) are fixed and |c|/μ < 1. This results from the opposite directions of shifts of the dimensionless E (to the left) and the stability threshold (to the right) with simultaneous compensating c-growth (∝ 1/ζ). It should be noted that the DS becomes unstable if |c|/μ > 1. If this occurs, additional growth of |β| is required. Physically, the ζ-decrease for a fixed SESAM can be provided at the expense of a larger mode size on it.

The decrease of γ (the b-decrease due to SPM suppression) enhances the DS stability if other parameters (including the dimensional energy E) are fixed. This effect results from a slower dependence of |c| on E in relation to that in the ADR. Physically, it can be realized by means of cavity purification for a thin-disk oscillator or by using a large-area fiber for an ANDi fiber laser. It should be noted here that such stability enhancement is not so substantial in relation to a Kerr-lens mode-locked oscillator obeying the cubic-quintic CNGLE [18,20], because there is no an asymptotic c = const for E → ∞ for the generalized CNGLE (Figs. 3, 5 cf. Fig. 6 in [18]).

Fig. 6 Thresholds (|c| ≡ τΣγ/|β|ζ for the solid-state oscillators and |c| ≡ τγ/|β|ζ for the fiber ones vs. E/b) of the pulse stability. The different colors correspond to b ≡ γ/ζ =33 and μ=0.5 (red), b =0.3 and μ=0.02 (blue), b =7×10⁻⁵ and μ=0.025 (black). Analytical thresholds from [20] correspond to the curves (the CDS is stable in the direction of the arrow marked in regular type). Analytical threshold from the variational model is shown by crosses (the CDS is stable in the direction of the arrows marked in italics). Symbols demonstrate the operational points in the vicinity of the stability threshold for the different oscillators (corresponding references and output energies are given).

Download Full Size | PDF

From the point of view of the master diagrams in Fig. 5, the almost ideal energy scalability corresponds to an ANDi fiber laser (if the spectral dissipation is defined by a spectral filter). Simple length scaling would result in proportional scaling of γ and |β|. Hence, the |c|-parameter remains constant. Simultaneously, the shift of the stability border to the right (due to b-growth) would provide the conditions for almost proportional energy growth (if |c| ≪ 1).

It should be noted that the dependence of the threshold curves on b (i.e. three-dimensionality of the master diagram) in the model under consideration is in completely agreement with the dependence of such a threshold on only the two parameters c and E in [15, 18] for |c| ≪ 1. Figure 5 suggests that the master diagram would become two-dimensional for |c| ≪ 1 if the energy is normalized to the value $\propto ζ^{2} / γ \sqrt{τ}$ . This means that the curves in Fig. 5 would merge (and the curves represented by symbols would merge for |c| ≪ 1) if one transits from E to E/b for fixed μ (see Fig. 6). Thus, this normalization in combination with the definition of c can be considered as the scaling law for the NDR in the |c| ≪ 1 limit.

Physically, the re-scaled master diagram can be divided into three main regions (Fig. 6): i) b ≪ 1 corresponds to a thin-disk solid-state oscillator (black curve and crosses; see Table 2 and black circle); ii) b > 1 corresponds to ANDi fiber oscillator (red curve; see Table III in [20] and red squares in Fig. 6); and iii) b < 1 corresponds to a broad-band solid-state oscillator (blue curve; see Table IV in [20] and blue triangle). One can see that the scaling rules under consideration are valid within an extremely broad range of parameters covering both solid-state (bulk and thin-disk) and fiber oscillators operating in the NDR. It should be noted that strong pulse breathing during its evolution in a fiber oscillator can deform the master diagram, and this effect calls for additional study.

5. Conclusion

The analytical theory of dissipative soliton (DS) energy scalability is presented. The theory is based on the variational approach to generalized CNGLE modeling of an oscillator mode-locked by a perfectly saturable absorber, e.g. a SESAM. The solutions presented cover two physically important cases: oscillators operating in the anomalous dispersion regime (ADR) and the normal dispersion regime (NDR).

It is found that the ADR is characterized by a two-dimensional master diagram connecting the DS stability with the relative contribution of dissipative and nondissipative effects for a given DS energy. Such a relative contribution is described by the sole parameter c ≡ τΣγ/βζ (or c ≡ τγ/βζ when the spectral dissipation is defined by a filter), which can easily be expressed through the values of the oscillator dispersion (β), self-phase modulation (γ), squared inverse gain bandwidth (τ), self-amplitude modulation (ζ), and net loss (Σ). Two different sectors of the ADR master diagram are found: one corresponding to a scaling pulse width and the other to an almost constant pulse width. For the latter, the c-parameter depends on the energy almost linearly. This means that the dispersion has to be increased, and/or the self-phase modulation has to be decreased, and/or the spectral filter bandwidth has to become broader with the growing energy. The approximated scaling rule for a dimensional energy in the constant pulse width sector can be formulated as

c E ζ / \sqrt{τ Σ \approx ℓ},

where ℓ grows almost linearly with the modulation depth parameter μ, and Σ appears only if the spectral dissipation is dominated by a gain-band (otherwise Σ =1).

The main results in the AND are verified by numerical simulations for thin-disk high-energy Yb:YAG oscillators. Moreover, it is demonstrated that a broad range of oscillators (both solid-state and fiber ones) fits to the model proposed.

The NDR providing CDS generation is characterized by a three-dimensional master diagram in the general case. The CDS is stabilized by substantially reduced (approximately tenfold) dispersion |β| in relation to that in the ADR. The already proposed approximated model [20] is in perfect agreement with the numerical simulation for thin-disk Yb:YAG oscillators. Simultaneously, comparison between the model presented and that of Ref. [20] demonstrates that the latter breaks when |c|/μ → 1. The approximated scaling rule for |c| ≪ 1, where the master diagram becomes two-dimensional, can be formulated as

E ζ^{2} c^{2} / γ \sqrt{τ Σ \approx ϒ,}

where ϒ is the growing function of μ slowly varying with c. This rule covers all types of oscillators. For solid-state oscillators, this scaling rule suggests a substantial reduction of dispersion and the need for SPM suppression in the NDR. For ANDi fiber oscillators, the scaling rule is almost perfect: fiber length scaling provides almost linear scaling of the CDS energy.

The main results in the NDR are verified numerically for the thin-disk Yb:YAG oscillators as well. It is demonstrated that the analytical model fits the real-world oscillators with parameters covering an extremely wide range.

Acknowledgments

This work was supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (FWF project P20293) and the Munich Centre for Advanced Photonics (MAP). The authors thank O. Pronin and R.Graf for valuable and stimulating discussions as well as for providing the simulation parameters for the Yb:YAG oscillators.

References and links

1. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006). [CrossRef]

2. T. Südmeyer, S. V. Marchese, C. R. E. Baer, G. Gingras, B. Witzel, and U. Keller, “Femtosecond laser oscillators for high-field science,” Nat. Photonics 2, 599–604 (2008). [CrossRef]

3. S. V. Marchese, C. R. E. Baer, A. G. Engqvist, S. Hashimoto, D. J. H. C. Maas, M. Golling, T. Südmeyer, and U. Keller, “Femtosecond thin disk laser oscillator with pulse energy beyond the 10-microjoule level,” Opt. Express 16, 6397–6407 (2008). [CrossRef] [PubMed]

4. J. Neuhaus, D. Bauer, J. Zhang, A. Killi, J. Kleinbauer, M. Kumkar, S. Weiler, M. Guina, D. H. Sutter, and Th. Dekorsy, “Subpicosecond thin-disk laser oscillator with pulse energies of up to 25.9 microjoules by use of an active multipass geometry,” Opt. Express 16, 20530–20539 (2008). [CrossRef] [PubMed]

5. G. Palmer, M. Schultze, M. Siegel, M. Emons, U. Bünting, and U. Morgner, “Passively mode-locked Yb:KLu(WO₄)₂ thin-disk oscillator operated in the positive and negative dispersion regime,” Opt. Lett. 33, 1608–1610 (2008). [CrossRef] [PubMed]

6. A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtoseond fiber laser,” Opt. Express 14, 10095–10100 (2006). [CrossRef] [PubMed]

7. S. Lefrançois, Kh. Kieu, Y. Deng, J. D. Kafka, and F. W. Wise, “Scaling of dissipative soliton fiber lasers to megawatt peak powers by use of large-area photonic crystal fiber,” Opt. Lett. 35, 1569–1571 (2010). [CrossRef] [PubMed]

8. S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005). [CrossRef]

9. E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr⁴⁺:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008). [CrossRef] [PubMed]

10. Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005). [CrossRef] [PubMed]

11. Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008). [CrossRef]

12. N. N. Akhmediev and A. Ankiewicz (Eds.), Dissipative Solitons: From Optics to Biology and Medicine. (Springer-Verlag, Berlin, Heidelberg, 2008).

13. E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005). [CrossRef]

14. V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005). [CrossRef]

15. V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B: Lasers Opt. 83, 503–510(2006). [CrossRef]

16. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008). [CrossRef]

17. W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008). [CrossRef]

18. V. L. Kalashnikov, “The unified theory of chirped-pulse oscillators,” Proc. SPIE Nonlinear Opt. Appl. III, Vol. 7354, Mario Bertolotti, Ed., p. 73540T (also arXiv:0903.5396 [physics.optics]) (2009).

19. V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E 80, 046606 (2009). [CrossRef]

20. V. L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: A unified standpoint,” Phys. Rev. A 79, 043829 (2009). [CrossRef]

21. W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009). [CrossRef]

22. Ph. Grelu, W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonance as a guideline for high-energy pulse laser oscillators,” J. Opt. Soc. Am. B 27, 2336–2341 (2010). [CrossRef]

23. D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear equations in optics,” Pramana J. Phys. 57, 917–936 (2001). [CrossRef]

24. J. G. Caputo, N. Flytzanis, and M. P. Sørensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139–145 (1995). [CrossRef]

25. S. Ch. Cerda, S. B. Cavalvanti, and J. M. Hickmann, “A variational approach of nonlinear dissipative pulse propagation,” Eur. Phys. J. D 1, 313–316 (1998). [CrossRef]

26. Ch. Jirauschek and F. X. Kärtner, “Gaussian pulse dynamics in gain media with Kerr nonlinearity,” J. Opt. Soc. Am. B 23, 1776–1784 (2006). [CrossRef]

27. C. Antonelli, J. Chen, and F. X. Kärtner, “Intracavity pulse dynamics and stability for passively mode-locked lasers,” Opt. Express 15, 5919–5924 (2007). [CrossRef] [PubMed]

28. B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008). [CrossRef]

29. N. N. Akhmediev and A. AnkiewiczSolitons: nonlinear pulses and beams (Chapman&Hall, London, 1997).

30. B. G. Bale, J. N. Kutz, A. Chong, W. H. Renninger, and F. W. Wise, “Spectral filtering for high-energy mode-locking in normal dispersion fiber lasers,” J. Opt. Soc. Am. B 25, 1763–1770 (2008). [CrossRef]

31. B. G. Bale, S. Boscolo, and S. K. Turitsyn, “Dissipative dispersion-managed solitons in mode-locked lasers,” Opt. Lett. 34, 3286–3288 (2009). [CrossRef] [PubMed]

32. H. A. Haus, “A theory of fast saturable absorber modelocking,” J. Appl. Phys. 46, 3049 (1975). [CrossRef]

33. H. A. Haus and Y. Silberberg, “Theory of mode locking of a laser diode with a multiple-quantum-well structure,” J. Opt. Soc. Am. B 2, 1237–1243 (1985). [CrossRef]

34. The Wolfram Mathematica 7 notebook is accessible at http://info.tuwien.ac.at/kalashnikov/variational.html

35. K. E. Oughstun, Electromagmetic and Optical Pulse Propagation 1 (Springer, NY, 2006).

36. R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B: Lasers Opt. 79, 163–173 (2004).

37. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003). [CrossRef]

38. F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992). [CrossRef]

39. A. K. Komarov and K. P. Komarov, “Pulse splitting in a passive mode-locked laser,” Optics Commun. 183, 265–270 (2000). [CrossRef]

40. S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr⁴⁺:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003). [CrossRef]

41. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18, 1732–1741 (2001). [CrossRef]

42. O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Optics Commun. 269, 156–165 (2007). [CrossRef]

43. A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25, 140–148 (2008). [CrossRef]

44. S. Lefrançois, K. Kieu, Y. Deng, J. D. Kafka, and F. W. Wise, “Scaling of dissipative soliton fiber lasers to megawatt peak powers by use of large-area photonic crystal fiber,” Opt. Lett. 35, 1569–1571 (2010). [CrossRef] [PubMed]

45. M. Siegel, G. Palmer, M. Emons, M. Schutze, A. Ruehl, and U. Morgner, “Pulsing dynamics in Ytterbium based chirped-pulse oscillators,” Opt. Express 16, 14314–14320 (2008). [CrossRef] [PubMed]

map	γ_air, GW⁻¹	γ_g, GW⁻¹	β_g, fs²	Ω_g, THz	σ⁻¹, mJ	ζ, MW⁻¹	κ
a	–	0.45	2050	5.3	3	2.25	0.07
b	3	0.12	520	5.3	0.24	0.71	0.072
b′	0.6	0.11	–	5.3	0.085	0.42	0.004–0.008

map	γ_air, GW⁻¹	γ_g, GW⁻¹	β_g, fs²	Ω_g, THz	σ⁻¹, mJ	ζ, MW⁻¹	κ
a	–	0.45	2050	5.3	3	2.25	0.07
b	3	0.12	520	5.3	0.24	0.71	0.072
b′	0.6	0.11	–	5.3	0.085	0.42	0.004–0.008

Energy scalability of mode-locked oscillators: a completely analytical approach to analysis

Abstract

1. Introduction

2. Variational DS solutions of the generalized CNGLE

3. Numerical DS solutions of the generalized CNGLE

4. Results and discussion

4.1. Anomalous dispersion regime (ADR)

4.2. Normal dispersion regime (NDR)

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (2)

Equations (14)

Optics Express

ADR	anomalous dispersion regime
NDR	normal dispersion regime
CPO	chirped-pulse oscillator
ANDi	all-normal-dispersion oscillator
DS	dissipative soliton
CDS	chirped dissipative soliton
CNGLE	complex nonlinear Ginzburg-Landau equation
SPM	self-phase modulation with coefficient γ
SAM	self-amplitude modulation with coefficient ζ
GDD	group-velocity dispersion with coefficient β (β < 0 corresponds to NDR)
τ	squared inverse gain (filter) bandwidth
γ	inverse SPM power
ζ	inverse loss saturation power
μ	saturable loss coefficent
Σ	net-loss coefficient
Ξ	saturated net-gain coefficient
E	pulse energy
T	pulse width
c ≡ τΣγ/\|β\|ζ	key control parameter defining the master diagram
b ≡ γ/ζ	parameter defining the relative contribution of SPM and SAM