DOI: 10.1007/s00340-006-2214-1 Appl. Phys. B 83, 503–510 (2006) Lasers and Optics Applied Physics B v.l. kalashnikov 1, ✉ e. podivilov 2 a. chernykh 2 a. apolonski 2,3 Chirped-pulse oscillators: theory and experiment 1 Institut für Photonik, TU Wien, Gusshausstr. 27/387, 1040 Vienna, Austria 2 Institute of Automation and Electrometry, RAS, 630090 Novosibirsk, Russia 3 Department für Physik der Ludwig-Maximilians-Universität München, Am Coulombwall 1, 85748 Garching, Germany Received: 18 January 2006/Revised version: 16 March 2006 Published online: 14 April 2006 • © Springer-Verlag 2006 ABSTRACT Theory of chirped-pulse oscillators operating in the positive dispersion regime is presented. It is found that the chirped pulses can be described analytically as solitary pulse solutions of the nonlinear cubic-quintic complex Ginzburg– Landau equation. Due to the closed form of the solution, basic characteristics of the regime under consideration are easily traceable. Numerical simulations validate the analytical tech- nique and the chirped-pulse stability. Experiments with 10 MHz Ti:Sa oscillator providing up to 150 nJ chirped pulses, which are compressible 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 pulses with energy exceeding 100 nJ are of interest for numer- ous applications, such as frequency conversion, frequency comb generation, micro-machining, etc. A well-known way to achieve the over-μ J femtosecond pulse energy is realized by oscillator-amplifier systems. However, such systems have high cost and comparatively low energy stability and pulse repetition rate. On the other hand, only ≈ 10 nJ femtosecond pulses are achievable directly from a femtosecond solid-state oscillator 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 such an oscillator up to 100 nJ [1, 2], but makes the system more complex. A promising approach to the μ J pulse energy frontier is based on an considerable decrease of the oscillator repeti- tion rate [3, 4]. However, such long-cavity oscillators suffer from strong instabilities caused by enhanced nonlinear effects within an active medium, which result from high intracavity power. It is possible to suppress instabilities by means of the pulse power lowering due to the pulse stretching. If the pulse is a soliton [5], such stretching requires a fair amount of the net negative group-delay dispersion (GDD; all abbreviations are summarized in Table 1) inside the oscillator cavity [6, 7]. This increases the pulse width, which cannot be reduced fur- ✉ Fax: +43-1-5880138799, E-mail: kalashnikov@tuwien.ac.at ther because the formed soliton is chirp-free. An alternative technique is to use an oscillator operating in the positive dis- persion regime (PDR) [8–11]. In PDR the chirped solitary pulse (CSP) develops. On the one hand, owing to its picosec- ond width, CSP has the peak power lower than the critical power of self-focusing P cr 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 exceed 10 15 W/cm 2 . This opens a way to low-cost, high-stable and compact light sources for highly-nonlinear physics, material processing, etc. PDR positive dispersion regime CSP chirped solitary pulse GDD group-delay dispersion KLM Kerr-lens mode-locking SPM self-phase modulation SAM self-amplitude modulation CGLE complex Ginzburg–Landau equation β GDD coefficient α square of the inverse gain bandwidth κ SAM parameter ζ parameter of SAM saturation γ SPM coefficient c ≡ αγ/βκ control parameter defining PDR σ net-loss coefficient A slowly varying field amplitude E spectral amplitude P(t) ( P(0)) intracavity instant (peak) power P cr critical power of self-focusing P av ( P out av ) intracavity (output) cw power (i.e. averaged power) p spectral power E (E out ) intracavity (output) pulse energy E ∗ energy stored inside the oscillator cavity in cw regime δ “stiffness coefficient” of the gain saturation T intracavity pulse duration T cav cavity period T r gain relaxation time ω frequency Ω instant frequency Δ parameter of the spectral truncation Ω L half-width of the Lorentzian spectral profile ψ ( Q) chirp (spectral chirp) χ output mirror transmission coefficient g (l) gain (loss) coefficient TABLE 1 Basic abbreviations and symbols (primed symbols correspond to normalized values)