1. Introduction

In the last decades, the breakthrough in the energy scalability of femtosecond laser pulses has been achieved that bring high-energy physics on tabletops of a mid-level university lab [1, 2]. As a result, the intensities of 1015 W/cm<sup>2</sup> become available directly from a mode-locked thin-disk laser oscillator operating at an over-MHz repetition rate [3–8]. Such systems are

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

considerably more simple, functional and economical than classical chirped-pulse amplifiers (of course, at the ~GW-pulse power level) [9, 10]. Moreover, a high repetition rate provides the signal/noise ratio improvement of 103 � 104 in comparison with an ordinary kHz chirped pulse amplifier. In practice, such oscillators are of interest for direct gas ionization and highharmonic generation [11, 12], pump-probe diffraction experiments with electrons [13] and fabrication of fine structures in transparent and semi-transparent materials [14], characterization and control of the electronic dynamics, metrology and ultra-sensitive spectroscopy, biophotonics and biomedicine, etc. [2, 15, 16].

of energy within the strongly confined spectral region. It is important that all excited and

In a trivial model of laser, there exists a set of N�longitudinal resonator modes which are separated by the frequency interval of δω ¼ 2π=Tres and excited by a gain medium with the gain-bandwidth Ω: N � Ω=δω [29]. These modes are mutually phase-independent and inco-

lar field beatings with the width of separated spikes � 1=Nδω ¼ 1=Ω (Figure 1(a); the intermode phase difference δϕ is random). However, a fixed inter-mode phase difference results in regular spikes of the � <sup>1</sup>=<sup>Ω</sup> width with the peak power � <sup>N</sup><sup>2</sup> and the repetition-period <sup>¼</sup> Tres (Figure 1(b)) [30]. The last phenomenon is called mode-locking (ML) and underlies a coherent energy condensation within short-time intervals. Respectively, the spectral width of each spike

However, this simple scheme faces many complications. Well, to be precise, a gain-band is not uniform (bell-shape like) and a mode, which is closest to a frequency ω<sup>0</sup> at gain maximum, has maximum amplification. Since laser gain is energy-saturable, this mode concentrates all energy and suppresses the competitive modes. This is a mode selection process. Therefore, a multimode generation leading to ultrashort pulse formation is not a genuine but emergent phenomenon

There are several possible mechanisms for such instability [31] which are closely connected with the issue of the ML self-start. Existing theories of the ML self-start predict a lot of effects involved in a laser pulse formation including mode-beatings [32, 33] and hole burning, induced refractive grating in an active medium [34], dynamic gain saturation [33, 35], parasitic reflections and absorption in a resonator [36], continuous-wave instability [37, 38] and Risken-Nummedal-Graham-Haken effect [39]. The thermodynamic theory of ML self-start has been developed, and it has been shown that the pulse appearance is a first-order phase transition, which is affected strongly by the laser noises distributed over a whole resonator period [40–42]. In any case, a stable ML requires whether a nonlinear resonant excitation by

<sup>n</sup>¼�N=<sup>2</sup> ai cos ð Þ <sup>ω</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup>δω <sup>t</sup> <sup>þ</sup> <sup>n</sup>δϕ � � is the irregu-

Theory of Laser Energy Harvesting at Femtosecond Scale http://dx.doi.org/10.5772/intechopen.75039 175

amplified modes must be phase-synchronized, i.e., coherent.

herent, and a result of their interference <sup>A</sup> <sup>¼</sup> <sup>P</sup><sup>N</sup>=<sup>2</sup>

which requires a multimode instability.

Figure 1. Interference of phase uncoupled (a) and locked (b) modes (N ¼ 20) [30].

tends to ΔΩ.

The phenomenon of ultrashort pulse energy harvesting exceeds the limits of immediate laserbased applications and is involved in a much broader context of formation and control of macroscopic coherent structures [17]. The high-energy ultrafast lasers become an excellent tool for testing the fundamental problems of self-organization and nonlinear dynamics far from thermodynamic equilibrium which cover the area ranging from hydrodynamics to condensed matter physics and even biology and sociology [18–20]. Such an approach based on the transfer of issue of complicate dynamics to another simpler material context can be named "metaphoric" or "analog" modeling [21, 22] and successes due to high controllability, relative simplicity, and unique potential of statistic gathering inherent in lasers systems [1].

The idea of energy E harvesting is based on an elementar relation: E ¼ PavTres, where Pav is an average power in a resonator with an effective period Tres. Scaling of Pav and/or Tres would provide the scaling of ultrashort pulse energy on condition that a stable ultrashort pulse emerges spontaneously (so-called, mode-locking self-start condition) in a laser system. As will be shown, these conditions are highly non-trivial for energy-scalable lasers and can limit substantially the pulse energy and its width. Two main approaches to the energy harvesting at femtosecond scale will be considered in this work.

The first one is based on the unique capacity of laser dissipative solitons (DS) [17, 23] to accumulate an energy without loss of stability [24, 25]. Some basic approaches to study of the energy-scaling laws for such systems will be presented, and the limits of energy and pulse width scalability will be outlined.

The second approach is based on the energy storing in an external high-Q resonator (so-called enhancement resonator, ER) coupled synchronously with a femtosecond pulse oscillator [26–28]. This simple idea faces difficulties when it is realized on a femtosecond scale because nonlinear effects and group-delay dispersion (GDD) tend to destroy a synchronization between a laser and ER. These issues will be outlined, and some modifications of ER technique will be proposed.
