2. DS energy scaling

The DS energy and width scaling are connected closely with a duality between amplification of the maximum number of laser modes and simultaneous spectral condensation, i.e., the concentration of energy within the strongly confined spectral region. It is important that all excited and amplified modes must be phase-synchronized, i.e., coherent.

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,

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

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

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

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 pro-

The DS energy and width scaling are connected closely with a duality between amplification of the maximum number of laser modes and simultaneous spectral condensation, i.e., the concentration

simplicity, and unique potential of statistic gathering inherent in lasers systems [1].

biophotonics and biomedicine, etc. [2, 15, 16].

174 High Power Laser Systems

scale will be considered in this work.

width scalability will be outlined.

2. DS energy scaling

posed.

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 incoherent, and a result of their interference <sup>A</sup> <sup>¼</sup> <sup>P</sup><sup>N</sup>=<sup>2</sup> <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 irregular 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 tends to ΔΩ.

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 which requires a multimode instability.

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

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

external periodic "force" (active mode-locking) <sup>1</sup> or a mutual mode coupling through an optical nonlinearity (passive mode-locking). The excitation of harmonics at �nδω couples and synchronizes the adjacent modes and provides their phase-locking.

<sup>4</sup>γα<sup>2</sup> <sup>þ</sup> <sup>6</sup>∂z<sup>ϕ</sup> <sup>¼</sup> <sup>β</sup>2=T<sup>2</sup>

Eq. (7) is the energy conservation law, and Eqs. (5) and (6) give the parameters of Schrödinger

where the first expression is the soliton area theorem, and the last one defines the soliton wave-

The effect of dissipative factors can be taken into account using the Ritz-Kantorovich method [51–54] when the reduced Euler-Lagrange equations are driven by a dissipative "source" [52]:

irð Þ a þ τ∂t,ta

parameters bi <sup>¼</sup> <sup>α</sup>; <sup>T</sup>; <sup>ϕ</sup> � �. The dissipative factors are defined by a net-loss with the coefficient Γ, a small-signal gain r with the inverse saturation energy σ and a squared inverse gain bandwidth τ. The self-amplitude modulation (SAM) providing ML is described by an effective "perfectly saturable absorber" [24] with the modulation depth μ, and the inverse saturation

A solution obtained from Eqs. (3) and (9) gives the expressions for area theorem and phase delay corresponding to the Schrödinger soliton. But the DS amplitude is defined by dissipative

<sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>p</sup> <sup>þ</sup> <sup>2</sup> <sup>Σ</sup> <sup>þ</sup> <sup>μ</sup> � � � <sup>2</sup>

It is convenient to assume that a laser operates in the vicinity of a threshold in steady-state

1 þ σ Ð j j a 2 dt þ

<sup>¼</sup> 2Re <sup>ð</sup>

<sup>Q</sup> <sup>∂</sup>a<sup>∗</sup> ∂bi dt,

> <sup>i</sup>μζj j <sup>a</sup> <sup>2</sup> a

1 þ ζj j a

3

2 :

Ldt is calculated using a trial function (i.e., Eq. (3)) with the

, q � �∂z<sup>ϕ</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup>

2α<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffi β2=γ q

> � ∂ Ð Ldt ∂bi

αT ¼

d dz ∂ Ð Ldt ∂ð Þ bi <sup>z</sup>

Q ¼ �iΓa þ

<sup>μ</sup> log <sup>1</sup>þα2�<sup>α</sup> ffiffiffiffiffiffiffiffi

α ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>β</sup>2<sup>ζ</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>α<sup>2</sup>T<sup>σ</sup> � ��<sup>1</sup>

β2 ζ.

<sup>1</sup>þα<sup>2</sup> <sup>p</sup> <sup>1</sup>þα2þ<sup>α</sup> ffiffiffiffiffiffiffiffi <sup>1</sup>þα<sup>2</sup> <sup>p</sup> � �

soliton if ∂zT ¼ 0, ∂zα ¼ 0:

2.1.1. Perfectly saturable absorber

Here the reduced Lagrangian Ð

number q.

power ζ.

factors [52]:

where <sup>Σ</sup> <sup>¼</sup> <sup>r</sup>

regime: <sup>r</sup>

<sup>1</sup>þ2α2T<sup>σ</sup> � <sup>Γ</sup>, C <sup>¼</sup> <sup>r</sup>τγ

<sup>1</sup>þ2α2T<sup>σ</sup> <sup>≈</sup> <sup>Γ</sup> and <sup>Σ</sup><sup>≈</sup> <sup>0</sup>, C<sup>≈</sup> <sup>Γ</sup>τγ

, (5)

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

<sup>2</sup>T<sup>2</sup> , (8)

<sup>C</sup>α<sup>2</sup> <sup>¼</sup> <sup>0</sup>, (10)

: The peak power α<sup>2</sup> in Eq. (10) is normalized to ζ.

(9)

177

<sup>β</sup>2=T<sup>2</sup> <sup>þ</sup> <sup>2</sup> γα<sup>2</sup> <sup>þ</sup> <sup>3</sup>∂z<sup>ϕ</sup> � � <sup>¼</sup> <sup>0</sup>, (6)

∂zT þ 4Tα∂zα ¼ 0: (7)

The mechanisms of ML are beyond the scopes of this work, and we will focus on the principles of the sustained ML energy-scalable regimes at femtosecond scale. The basic principle under consideration is to exploit DS [23] which is extremely stable in nonequilibrium dissipative environment [24, 25]. Since DS behaves like a soliton of integrable systems [17, 44], its dynamics can be described by some distributed nonlinear model. The most famous and studied one is based on the complex nonlinear Ginzburg-Landau equation which can be treated as a dissipative extension of the nonlinear Schrödinger equation [45].<sup>2</sup>

A very productive approach to the study of this class of equations is based on the so-called variational approximation (VA) [49–51]. The non-dissipative effects can be described by the Lagrangian density:

$$L = \frac{i}{2} \left( a^\*(z, t) \partial\_z a(z, t) - a(z, t) \partial\_z a^\*(z, t) \right) - \frac{1}{2} \gamma |a(z, t)|^4 + \frac{\beta\_2}{2} \partial\_t a(z, t) \partial\_t a^\*(z, t), \tag{1}$$

where a zð Þ ; t is a complex slowly varying field amplitude, t and z are local time and propagation distance, respectively, γ is a self-phase modulation (SPM) coefficient and β<sup>2</sup> is a groupdelay dispersion (GDD) coefficient. The Euler-Lagrange equation corresponding to Eq. (1) is the nonlinear Schrödinger equation:

$$d\partial\_z a(z,t) = \frac{\beta\_2}{2} \partial\_{t,t} a(z,t) + \gamma |a(z,t)|^2 a(z,t). \tag{2}$$

Further, two different types of DS will be considered: (i) chirped-free

$$a(z,t) = a(z)\text{sech}(t/T(z))\exp\left(i\phi(z)\right),\tag{3}$$

and (ii) chirped pulses

$$a(z,t) = a(z) \text{sech}(t/T(z))^{1+i\psi(z)} \exp\left(i\phi(z)\right),\tag{4}$$

where α, Τ, ψ and ϕ are DS amplitude, width, chirp, and phase-delay, respectively.

### 2.1. Chirped-free DS

VA in action looks like following. Substitution of the trial solution (3) into (1) with the subsequent integration over t results in the reduced Euler-Lagrange equations [52]:

<sup>1</sup> This phenomenon is closely related to the concept of stochastic resonance which describes processes of resonant coherence enhancement in a noisy periodically driven system [43].

<sup>2</sup> Different versions of this equation describe an extremely broad area of phenomena ranging from laser dynamics [17, 46, 47], oscillatory chemical reactions [22] to Bose-Einstein condensations and biological systems [48].

$$4\gamma a^2 + 6\partial\_z \phi = \beta\_2 / T^2,\tag{5}$$

$$2\beta\_2/T^2 + 2\left(\gamma\alpha^2 + 3\partial\_z\phi\right) = 0,\tag{6}$$

$$2\alpha^2 \partial\_z T + 4Ta \partial\_z \alpha = 0.\tag{7}$$

Eq. (7) is the energy conservation law, and Eqs. (5) and (6) give the parameters of Schrödinger soliton if ∂zT ¼ 0, ∂zα ¼ 0:

$$
\alpha T = \sqrt{\beta\_2/\gamma}, \ q \equiv -\mathfrak{d}\_z \phi = \frac{\beta\_2}{2T^2}, \tag{8}
$$

where the first expression is the soliton area theorem, and the last one defines the soliton wavenumber q.

#### 2.1.1. Perfectly saturable absorber

external periodic "force" (active mode-locking)

Lagrangian density:

176 High Power Laser Systems

<sup>L</sup> <sup>¼</sup> <sup>i</sup>

and (ii) chirped pulses

2.1. Chirped-free DS

enhancement in a noisy periodically driven system [43].

1

2

the nonlinear Schrödinger equation:

chronizes the adjacent modes and provides their phase-locking.

tive extension of the nonlinear Schrödinger equation [45].<sup>2</sup>

<sup>2</sup> <sup>a</sup><sup>∗</sup>ð Þ <sup>z</sup>; <sup>t</sup> <sup>∂</sup>za zð Þ� ; <sup>t</sup> a zð Þ ; <sup>t</sup> <sup>∂</sup>za<sup>∗</sup> <sup>ð</sup> ð Þ <sup>z</sup>; <sup>t</sup> Þ � <sup>1</sup>

<sup>i</sup>∂za zð Þ¼ ; <sup>t</sup> <sup>β</sup><sup>2</sup>

Further, two different types of DS will be considered: (i) chirped-free

where α, Τ, ψ and ϕ are DS amplitude, width, chirp, and phase-delay, respectively.

quent integration over t results in the reduced Euler-Lagrange equations [52]:

oscillatory chemical reactions [22] to Bose-Einstein condensations and biological systems [48].

VA in action looks like following. Substitution of the trial solution (3) into (1) with the subse-

This phenomenon is closely related to the concept of stochastic resonance which describes processes of resonant coherence

Different versions of this equation describe an extremely broad area of phenomena ranging from laser dynamics [17, 46, 47],

<sup>1</sup> or a mutual mode coupling through an optical

<sup>2</sup> <sup>∂</sup>ta zð Þ ; <sup>t</sup> <sup>∂</sup>ta<sup>∗</sup>ð Þ <sup>z</sup>; <sup>t</sup> , (1)

a zð Þ ; t : (2)

nonlinearity (passive mode-locking). The excitation of harmonics at �nδω couples and syn-

The mechanisms of ML are beyond the scopes of this work, and we will focus on the principles of the sustained ML energy-scalable regimes at femtosecond scale. The basic principle under consideration is to exploit DS [23] which is extremely stable in nonequilibrium dissipative environment [24, 25]. Since DS behaves like a soliton of integrable systems [17, 44], its dynamics can be described by some distributed nonlinear model. The most famous and studied one is based on the complex nonlinear Ginzburg-Landau equation which can be treated as a dissipa-

A very productive approach to the study of this class of equations is based on the so-called variational approximation (VA) [49–51]. The non-dissipative effects can be described by the

where a zð Þ ; t is a complex slowly varying field amplitude, t and z are local time and propagation distance, respectively, γ is a self-phase modulation (SPM) coefficient and β<sup>2</sup> is a groupdelay dispersion (GDD) coefficient. The Euler-Lagrange equation corresponding to Eq. (1) is

<sup>2</sup> <sup>∂</sup>t,ta zð Þþ ; <sup>t</sup> <sup>γ</sup>j j a zð Þ ; <sup>t</sup> <sup>2</sup>

2

<sup>γ</sup>j j a zð Þ ; <sup>t</sup> <sup>4</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup>

a zð Þ¼ ; <sup>t</sup> <sup>α</sup>ð Þ<sup>z</sup> sechð Þ <sup>t</sup>=T zð Þ exp <sup>i</sup>ϕð Þ<sup>z</sup> , (3)

a zð Þ¼ ; <sup>t</sup> <sup>α</sup>ð Þ<sup>z</sup> sechð Þ <sup>t</sup>=T zð Þ <sup>1</sup>þiψð Þ<sup>z</sup> exp <sup>i</sup>ϕð Þ<sup>z</sup> , (4)

The effect of dissipative factors can be taken into account using the Ritz-Kantorovich method [51–54] when the reduced Euler-Lagrange equations are driven by a dissipative "source" [52]:

$$\begin{split} \frac{d}{dz} \frac{\partial \int Ldt}{\partial (b\_i)\_z} - \frac{\partial \int Ldt}{\partial b\_i} &= 2 \text{Re} \int Q \frac{\partial a^\*}{\partial b\_i} dt, \\ Q = -i\Gamma a + \frac{i\rho(a + \tau \,\partial\_{t,\boldsymbol{\theta}} a)}{1 + \sigma \int |a|^2 dt} + \frac{i\mu \zeta |a|^2 a}{1 + \zeta |a|^2}. \end{split} \tag{9}$$

Here the reduced Lagrangian Ð Ldt is calculated using a trial function (i.e., Eq. (3)) with the parameters bi <sup>¼</sup> <sup>α</sup>; <sup>T</sup>; <sup>ϕ</sup> � �. The dissipative factors are defined by a net-loss with the coefficient Γ, a small-signal gain r with the inverse saturation energy σ and a squared inverse gain bandwidth τ. The self-amplitude modulation (SAM) providing ML is described by an effective "perfectly saturable absorber" [24] with the modulation depth μ, and the inverse saturation power ζ.

A solution obtained from Eqs. (3) and (9) gives the expressions for area theorem and phase delay corresponding to the Schrödinger soliton. But the DS amplitude is defined by dissipative factors [52]:

$$\frac{\mu \log \left( \frac{1 + a^2 - a\sqrt{1 + a^2}}{1 + a^2 + a\sqrt{1 + a^2}} \right)}{a\sqrt{1 + a^2}} + 2\left(\Sigma + \mu\right) - \frac{2}{3}\mathbb{C}\alpha^2 = 0,\tag{10}$$

where <sup>Σ</sup> <sup>¼</sup> <sup>r</sup> <sup>1</sup>þ2α2T<sup>σ</sup> � <sup>Γ</sup>, C <sup>¼</sup> <sup>r</sup>τγ <sup>β</sup>2<sup>ζ</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>α<sup>2</sup>T<sup>σ</sup> � ��<sup>1</sup> : The peak power α<sup>2</sup> in Eq. (10) is normalized to ζ. It is convenient to assume that a laser operates in the vicinity of a threshold in steady-state regime: <sup>r</sup> <sup>1</sup>þ2α2T<sup>σ</sup> <sup>≈</sup> <sup>Γ</sup> and <sup>Σ</sup><sup>≈</sup> <sup>0</sup>, C<sup>≈</sup> <sup>Γ</sup>τγ β2 ζ.

The marginal stability condition Σ ¼ 0 defines a stability of DS against continuous-wave or multiple pulse generation [30, 55], and the DS approaching this stability threshold has a minimum width and a best asymptotical energy scalability (<sup>E</sup> <sup>¼</sup> <sup>2</sup>α<sup>2</sup>Tζ<sup>=</sup> ffiffiffiffiffiffi <sup>τ</sup><sup>Γ</sup> <sup>p</sup> <sup>≫</sup> 1) [52]:

$$\begin{aligned} \mathbb{C} &\to \text{const} \times \sqrt{\mu \tau \Gamma} / \mathbb{E} \mathbb{Z}, \mathbb{E} \to \text{const} \times \frac{\beta\_2}{\gamma'} \sqrt{\frac{\mu}{\tau \Gamma}}, \\\alpha^2 &\to \left(\frac{\text{const}}{2}\right)^2 \frac{\mu \beta\_2}{\tau \Gamma \gamma'}, T \to \frac{2}{\text{const}} \sqrt{\frac{\tau \Gamma}{\mu}}, \text{const} \approx 3.535. \end{aligned} \tag{11}$$

Properties of DS are described by the so-called master diagram [24] which represents the DS parametric space and is shown in Figure 4(a). There are two DS-solutions of Eqs. (1), (3), (9) and (10): (i) upper branch (i.e., the branch with the larger C for fixed E and Σ, see Figure 4(a)) corresponds to the above considered energy-scalable DS. The energy scalability for this type of soliton is accompanied by minimization of its width (Figures 3 and 4(b); i.e., lim

Figure 3. Dependence of the normalized TFWHM on Σ (a) for μ = 0.05 (1), 0.1 (2), 0.15 (3), C ¼ 0:01; and on the normalized

has a threshold of marginal stability Σ =0(Figure 4(a); curve 1). (ii) lower branch (Figure 4 (a)) corresponds to a DS energy scalability provided by its width growth (Figure 4(b);

chirp-free pulse would require an additional nonlinear mechanism for external compression.

A fundamental property of the DS solutions presented is their stability. The Vakhitov-Kolokolov stability criterion dE=dq > 0 [56, 57] demonstrates the stability of both branches of

Figure 4. Master diagram (a) and the corresponding DS widths (b). Σ = 0 (1), �0.01 (2), and �0.02 (3), μ = 0.05.

, see Eq. (11)) and, respectively, by the growth of peak power. Namely, this branch

T ¼ ∞). Thus, this DS branch is unpractical for energy scaling because the broad

2 const

lim <sup>E</sup>!<sup>∞</sup>,Σ<<sup>0</sup>

ffiffiffiffi τ Γ μ q

energy E (b) for μ = 0.05, Σ = 0.

DS (see Eq. (8)):

One can name this branch as energy-unscalable.

<sup>E</sup>!<sup>∞</sup>,<sup>Σ</sup>!<sup>0</sup>

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

T ¼

179

These scaling laws demonstrate main principles of chirped-free pulse energy harvesting. Of course, these dependencies can be considered as only qualitative ones. Nevertheless, they demonstrate that the asymptotic DS energy scales ∝ β2, and the minimum pulse width is defined by not only the medium gain bandwidth ∝ 1= ffiffiffi <sup>τ</sup> <sup>p</sup> but the net-loss <sup>Γ</sup> and the modulation depth μ, as well (Figure 2).

Thus, the pulse can be squeezed by scaling of modulation depth with a parallel decrease of the stabilizing dispersion. Additional pulse shortening can be provided by net-loss lowering (see Figure 2). These tendencies are quite reasonable because the selective spectral properties of an active medium are defined by not the gain for a small signal but by the saturated gain ≈ Γ near the pulse stability threshold (i.e., since no gain, no gain induced spectral selection). Simultaneously, the modulation depth defines an inter-mode coupling strength that favors ML and, thereby, pulse spectrum broadening with μ-growth.

Eqs. (8) and (10) demonstrate that an approach to the threshold C (Σ ! 0) as well as a higher E∝ζ= ffiffiffi <sup>τ</sup> <sup>p</sup> minimize the pulse width <sup>T</sup> (Figure 3).

Figure 2. Dependence of the asymptotic full width at half maximum (FWHM) TFWHM on the modulation depth μ for different net-loss coefficients Γ. The gain bandwidth of 5.3 THz corresponds to a Yb: YAG thin-disk active medium.

The marginal stability condition Σ ¼ 0 defines a stability of DS against continuous-wave or multiple pulse generation [30, 55], and the DS approaching this stability threshold has a

μτ<sup>Γ</sup> <sup>p</sup> <sup>=</sup>Eζ, E ! const � <sup>β</sup><sup>2</sup>

, T ! <sup>2</sup> const

These scaling laws demonstrate main principles of chirped-free pulse energy harvesting. Of course, these dependencies can be considered as only qualitative ones. Nevertheless, they demonstrate that the asymptotic DS energy scales ∝ β2, and the minimum pulse width is

Thus, the pulse can be squeezed by scaling of modulation depth with a parallel decrease of the stabilizing dispersion. Additional pulse shortening can be provided by net-loss lowering (see Figure 2). These tendencies are quite reasonable because the selective spectral properties of an active medium are defined by not the gain for a small signal but by the saturated gain ≈ Γ near the pulse stability threshold (i.e., since no gain, no gain induced spectral selection). Simultaneously, the modulation depth defines an inter-mode coupling strength that favors ML and,

Eqs. (8) and (10) demonstrate that an approach to the threshold C (Σ ! 0) as well as a higher

Figure 2. Dependence of the asymptotic full width at half maximum (FWHM) TFWHM on the modulation depth μ for different net-loss coefficients Γ. The gain bandwidth of 5.3 THz corresponds to a Yb: YAG thin-disk active medium.

γ

, const ≈ 3:535:

ffiffiffiffiffiffi τΓ μ

s

ffiffiffiffiffiffi μ τΓ r , <sup>τ</sup><sup>Γ</sup> <sup>p</sup> <sup>≫</sup> 1) [52]:

<sup>τ</sup> <sup>p</sup> but the net-loss <sup>Γ</sup> and the modula-

(11)

minimum width and a best asymptotical energy scalability (<sup>E</sup> <sup>¼</sup> <sup>2</sup>α<sup>2</sup>Tζ<sup>=</sup> ffiffiffiffiffiffi

<sup>C</sup> ! const � ffiffiffiffiffiffiffiffiffi

<sup>α</sup><sup>2</sup> ! const 2 � �<sup>2</sup> μβ<sup>2</sup> τΓγ

defined by not only the medium gain bandwidth ∝ 1= ffiffiffi

thereby, pulse spectrum broadening with μ-growth.

<sup>τ</sup> <sup>p</sup> minimize the pulse width <sup>T</sup> (Figure 3).

tion depth μ, as well (Figure 2).

178 High Power Laser Systems

E∝ζ= ffiffiffi

Figure 3. Dependence of the normalized TFWHM on Σ (a) for μ = 0.05 (1), 0.1 (2), 0.15 (3), C ¼ 0:01; and on the normalized energy E (b) for μ = 0.05, Σ = 0.

Properties of DS are described by the so-called master diagram [24] which represents the DS parametric space and is shown in Figure 4(a). There are two DS-solutions of Eqs. (1), (3), (9) and (10): (i) upper branch (i.e., the branch with the larger C for fixed E and Σ, see Figure 4(a)) corresponds to the above considered energy-scalable DS. The energy scalability for this type of soliton is accompanied by minimization of its width (Figures 3 and 4(b); i.e., lim <sup>E</sup>!<sup>∞</sup>,<sup>Σ</sup>!<sup>0</sup> T ¼

2 const ffiffiffiffi τ Γ μ q , see Eq. (11)) and, respectively, by the growth of peak power. Namely, this branch has a threshold of marginal stability Σ =0(Figure 4(a); curve 1). (ii) lower branch (Figure 4 (a)) corresponds to a DS energy scalability provided by its width growth (Figure 4(b); lim <sup>E</sup>!<sup>∞</sup>,Σ<<sup>0</sup> T ¼ ∞). Thus, this DS branch is unpractical for energy scaling because the broad chirp-free pulse would require an additional nonlinear mechanism for external compression. One can name this branch as energy-unscalable.

A fundamental property of the DS solutions presented is their stability. The Vakhitov-Kolokolov stability criterion dE=dq > 0 [56, 57] demonstrates the stability of both branches of DS (see Eq. (8)):

Figure 4. Master diagram (a) and the corresponding DS widths (b). Σ = 0 (1), �0.01 (2), and �0.02 (3), μ = 0.05.

Figure 5. The dimensionless DS widths (a) and the corresponding net-gain (b) in dependence on the threshold energy Ξ for <sup>δ</sup> <sup>=</sup> �0.05, <sup>C</sup> = 10�<sup>3</sup> (1); <sup>δ</sup> <sup>=</sup> �0.5, <sup>C</sup> = 10�<sup>3</sup> (2); and <sup>δ</sup> <sup>=</sup> �0.05, <sup>C</sup> = 10�<sup>2</sup> (3). <sup>μ</sup> = 0.07.

$$\frac{d\mathbb{E}}{dq} = \frac{d}{dq}\left(2a^2T\right) = \frac{d}{dq}\left(2\sqrt{2\beta\_2q}/\gamma\right) = \frac{\sqrt{2\beta\_2/q}}{\gamma} > 0. \tag{12}$$

The solutions for T and q correspond to Eq. (8), but the two-branch solution for DS peak power

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup> � <sup>C</sup> <sup>2</sup> <sup>þ</sup> <sup>96</sup>ζΣ=5<sup>κ</sup>

κ=ζτΓ p and ffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffi 5β<sup>2</sup> ζγ s , (15)

181

(16)

.

κζ=τΓ p , respectively, and

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

5 2ð Þ � <sup>C</sup> <sup>=</sup><sup>C</sup> <sup>p</sup> <sup>=</sup>2 (curve 1

� � q

ffiffiffiffiffiffiffiffiffiffiffiffi 20μβ<sup>2</sup> κ γ s

ffiffiffiffiffiffiffiffi κβ<sup>2</sup> 5γμ s :

One can see that the energy scaling is provided by the DS width (not power) scaling that is a natural consequence of the peak power confinement <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>=2<sup>ζ</sup> imposed by a SAM saturation<sup>3</sup>

Most promising devices realizing the femtosecond-pulse energy scalability are thin-disk solidstate lasers [5, 6, 24, 58] which provide an excellent average power scaling and controllable nonlinear effects limiting the energy scalability in fiber oscillators [25]. Nevertheless, there are some main obstacles for further energy harvesting at femtosecond scale for such a type of

Figure 6. Master diagram for the chirp-free DS and the cubic-quintic SAM. Curve 1 divides two different branches of DS.

In the case of perfectly saturable SAM, the confinement is imposed by spectral dissipation (i.e., the DS width but not

5=C p corresponds to asymptotical energy scaling law.

¼

can be expressed in an explicit form:

in Figure 6), and are shown in Figure 6.

The upper line <sup>E</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi

power is confined; see Eq. (11)).

3

<sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>5</sup>

(power, time and energy are normalized to ζ, ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>C</sup> ! <sup>5</sup>τ<sup>Γ</sup> 4μζ<sup>2</sup>

<sup>α</sup><sup>2</sup> ! <sup>5</sup><sup>μ</sup> κ , T !

Both branches behave quite congruently in this case (Figure 6).

2.1.3. Energy harvesting of the chirp-free DS at femtosecond scale

The asymptotic scaling laws for this type of SAM are:

<sup>16</sup> <sup>2</sup> � <sup>C</sup><sup>∓</sup>

<sup>C</sup> <sup>¼</sup> <sup>τ</sup>Γγ=β2κ). These branches are separated by the energy curve <sup>E</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>E</sup><sup>2</sup> , E !

It means physically that an energy scalability of DS does not suffer from soliton collapse and both DS branches are feasible.

As was mentioned above, the net-gain Σ is energy-dependent, and such a dependence has to be taken into account. In the neighborhood of the laser threshold where Σð Þ E ≈ 0, one may expand the net-gain coefficient near a threshold energy <sup>Ξ</sup> <sup>¼</sup> <sup>σ</sup>�<sup>1</sup>ð Þ <sup>r</sup>=<sup>Γ</sup> � <sup>1</sup> :

$$
\Sigma = \frac{d}{dE} \left( \frac{\rho}{1 + \sigma E} - \Gamma \right) \Big|\_{E=\Xi} + O \left( \frac{d^2}{dE^2} \right) \approx \delta \left( \frac{E}{\Xi} - 1 \right) \tag{13}
$$

where <sup>δ</sup> ¼ �Γ<sup>2</sup> Ξσ=r

Figure 5(a) shows the DS dependence on the threshold energy for a fixed control parameter C in the presence of gain saturation. DS squeezes with energy, and such a squeezing is confined by the stability criterion Σ < 0. Simultaneously, Σ decreases from 0 with energy (Figure 5(b)), that corresponds to the "energy unscalable" DS, with the subsequent growth up to 0, that corresponds to the "energy scalable" DS. Thus, there is not a "switch" between two different types of DS in a real-world laser system which behaves quite smoothly with energy.

#### 2.1.2. Cubic-quintic SAM

Physically, this type of SAM describes approximately an action of nonlinear polarization rotation, which is a typical ML mechanism for fiber lasers, or a so-called "soft aperture" Kerrlens ML typical for solid-state lasers [24, 25]. In this case, loss saturation switches to the loss growth at <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>=2<sup>ζ</sup> and

$$Q = -i\Gamma a + \frac{\mathrm{i}\rho(a + \tau \,\partial\_{t,t}a)}{1 + \sigma \int |a|^2 dt} + i\kappa \left(1 - \zeta |a|^2\right) |a|^2 a. \tag{14}$$

The κ-parameter plays a role of the inverse loss saturation power, and the modulation depth is μ ¼ κ=4ζ.

The solutions for T and q correspond to Eq. (8), but the two-branch solution for DS peak power can be expressed in an explicit form:

$$a^2 = \frac{5}{16} \left( 2 - \mathbb{C} \mp \sqrt{\left( 2 - \mathbb{C} \right)^2 + 96 \zeta \Sigma / 5 \kappa} \right) \tag{15}$$

(power, time and energy are normalized to ζ, ffiffiffiffiffiffiffiffiffiffiffiffiffi κ=ζτΓ p and ffiffiffiffiffiffiffiffiffiffiffiffiffi κζ=τΓ p , respectively, and <sup>C</sup> <sup>¼</sup> <sup>τ</sup>Γγ=β2κ). These branches are separated by the energy curve <sup>E</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2ð Þ � <sup>C</sup> <sup>=</sup><sup>C</sup> <sup>p</sup> <sup>=</sup>2 (curve 1 in Figure 6), and are shown in Figure 6.

The asymptotic scaling laws for this type of SAM are:

dE dq <sup>¼</sup> <sup>d</sup>

<sup>Σ</sup> <sup>¼</sup> <sup>d</sup> dE

both DS branches are feasible.

Ξσ=r

where <sup>δ</sup> ¼ �Γ<sup>2</sup>

180 High Power Laser Systems

2.1.2. Cubic-quintic SAM

growth at <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>=2<sup>ζ</sup> and

μ ¼ κ=4ζ.

dq <sup>2</sup>α<sup>2</sup>

for <sup>δ</sup> <sup>=</sup> �0.05, <sup>C</sup> = 10�<sup>3</sup> (1); <sup>δ</sup> <sup>=</sup> �0.5, <sup>C</sup> = 10�<sup>3</sup> (2); and <sup>δ</sup> <sup>=</sup> �0.05, <sup>C</sup> = 10�<sup>2</sup> (3). <sup>μ</sup> = 0.07.

<sup>T</sup> � � <sup>¼</sup> <sup>d</sup>

expand the net-gain coefficient near a threshold energy <sup>Ξ</sup> <sup>¼</sup> <sup>σ</sup>�<sup>1</sup>ð Þ <sup>r</sup>=<sup>Γ</sup> � <sup>1</sup> :

r <sup>1</sup> <sup>þ</sup> <sup>σ</sup><sup>E</sup> � <sup>Γ</sup> � ��

Q ¼ �iΓa þ

dq <sup>2</sup> ffiffiffiffiffiffiffiffiffi <sup>2</sup>β2<sup>q</sup> <sup>p</sup> <sup>=</sup><sup>γ</sup> � � <sup>¼</sup>

Figure 5. The dimensionless DS widths (a) and the corresponding net-gain (b) in dependence on the threshold energy Ξ

It means physically that an energy scalability of DS does not suffer from soliton collapse and

As was mentioned above, the net-gain Σ is energy-dependent, and such a dependence has to be taken into account. In the neighborhood of the laser threshold where Σð Þ E ≈ 0, one may

Figure 5(a) shows the DS dependence on the threshold energy for a fixed control parameter C in the presence of gain saturation. DS squeezes with energy, and such a squeezing is confined by the stability criterion Σ < 0. Simultaneously, Σ decreases from 0 with energy (Figure 5(b)), that corresponds to the "energy unscalable" DS, with the subsequent growth up to 0, that corresponds to the "energy scalable" DS. Thus, there is not a "switch" between two different

Physically, this type of SAM describes approximately an action of nonlinear polarization rotation, which is a typical ML mechanism for fiber lasers, or a so-called "soft aperture" Kerrlens ML typical for solid-state lasers [24, 25]. In this case, loss saturation switches to the loss

The κ-parameter plays a role of the inverse loss saturation power, and the modulation depth is

<sup>þ</sup> <sup>i</sup><sup>κ</sup> <sup>1</sup> � <sup>ζ</sup>j j <sup>a</sup> <sup>2</sup> � �

j j a 2

<sup>þ</sup> <sup>O</sup> <sup>d</sup><sup>2</sup> dE<sup>2</sup> � �

� � � E¼Ξ

types of DS in a real-world laser system which behaves quite smoothly with energy.

irð Þ a þ τ∂t,ta

1 þ σ Ð j j a 2 dt

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>β2=<sup>q</sup> <sup>p</sup> γ

<sup>≈</sup> <sup>δ</sup> <sup>E</sup> <sup>Ξ</sup> � <sup>1</sup> � �

> 0: (12)

, (13)

a: (14)

$$\begin{aligned} \mathbb{C} &\to \frac{5\pi\Gamma}{4\mu\zeta^2 E^2}, \mathbb{E} \to \sqrt{\frac{20\mu\beta\_2}{\kappa\gamma}} = \sqrt{\frac{5\beta\_2}{\zeta\chi'}}\\ \alpha^2 &\to \frac{5\mu}{\kappa}, \mathbb{T} \to \sqrt{\frac{\kappa\beta\_2}{5\gamma\mu}}. \end{aligned} \tag{16}$$

One can see that the energy scaling is provided by the DS width (not power) scaling that is a natural consequence of the peak power confinement <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>=2<sup>ζ</sup> imposed by a SAM saturation<sup>3</sup> . Both branches behave quite congruently in this case (Figure 6).

#### 2.1.3. Energy harvesting of the chirp-free DS at femtosecond scale

Most promising devices realizing the femtosecond-pulse energy scalability are thin-disk solidstate lasers [5, 6, 24, 58] which provide an excellent average power scaling and controllable nonlinear effects limiting the energy scalability in fiber oscillators [25]. Nevertheless, there are some main obstacles for further energy harvesting at femtosecond scale for such a type of

Figure 6. Master diagram for the chirp-free DS and the cubic-quintic SAM. Curve 1 divides two different branches of DS. The upper line <sup>E</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi 5=C p corresponds to asymptotical energy scaling law.

<sup>3</sup> In the case of perfectly saturable SAM, the confinement is imposed by spectral dissipation (i.e., the DS width but not power is confined; see Eq. (11)).

devices. (i) a traditional ML mechanism uses the structured semiconductor devices (so-called semiconductor saturable absorber mirrors, SESAM) which (α) have a slow (~100 fs) response time; (β) a complicated and hardly-controllable kinematics including higher-order nonlinear effects, non-saturable losses, temperature and radiational damage, etc.; (γ) SAM and SPM effects are decoupled for this type of ML that requires growth of GDD for DS stabilization in accordance with the area theorem (Eq. (8)). (ii) As was shown (Eq. (11)), the minimum pulse width is ∝ ffiffiffiffiffiffiffiffiffiffiffi τΓ=μ p so that using the media with the broader gain band would provide a pulse shortening down to sub-100 fs [59]. However, such media with good optical quality are not widely available and technologically advanced.

Nevertheless, an alternative approach to energy scalability of femtosecond pulses has been demonstrated in [7]. (i) ML mechanism can be provided by an instantaneous self-focusing (Kerr-lensing) induced by a set of nonlinear crystals inside a laser resonator. (ii) Such a mechanism combines both SAM and SPM that enhances the SAM parameters (μ and ζ) in parallel with the SPM one (γ, see Eq. (11)). As a result, the GDD value can be reduced in parallel with the DS shortening in agreement with the area theorem. (iii) A real-world gain band profile is Lorentzian, not Gaussian as in Eqs. (9) and (14).

The Lorentzian gain profile can be taken into account by using the numerical simulations of the generalized complex nonlinear Ginzburg-Landau equation [60, 61]:

$$\partial\_z a = -\Gamma a + \frac{\mu \zeta |a|^2}{1 + \zeta |a|^2} a - i \left(\frac{\beta\_2}{2} \partial\_{t,t} a + \gamma |a|^2 a\right) + \frac{\rho \Omega}{2 \left(1 + \sigma \int |a|^2 dt\right)} \int\_{-\infty}^t a \left(z, t\right) \exp\left[-\Omega \left(t - t'\right)\right] d\zeta,\tag{17}$$

where a characteristic gain bandwidth is Ω ∝ 1= ffiffiffi <sup>τ</sup> <sup>p</sup> .

As was shown in [60], the Lorentzian gain profile gives more efficient amplification and broader spectrum than the Gaussian one. Additionally, an inherent gain dispersion shifts the DS spectrum and affects its shape [62]. The numerically obtained pulse spectra for different modulation depths are shown in Figure 7. One can see a pronounced spectrum broadening and, correspondingly, pulse shortening with the modulation depth growth.

The dependences of minimum DS width and corresponding stabilizing GDD on the modulation depth for different values of the SAM saturation power ζ are shown in Figure 8. One can see that the DS shortens with μ in qualitative agreement with the analytical results presented above. Simultaneously, the DS spectrum is noticeably broader than the gain band that provides a generation of sub-50 fs pulses at the MW peak power level directly from an oscillator. Since a pulse is chirp-free, the threshold stabilizing anomalous GDD decreases, as well, in agreement with the soliton area theorem. Simultaneously, there is the nonmonotonic dependence of DS width on the SAM parameter ζ so that T decreases initially and then increases with the ζ-decrease, i.e., with the saturation power growth. The growth of saturation power (ζ-decrease) causes a threshold-like increase of pulse width and stabilizing GDD for small modulation depths μ.

Figure 9 illustrates the energy-dependence of the DS width and the stabilizing GDD for a low net-loss. The short pulses are possible in this case, as well, but a small modulation depth does

Figure 8. Pulse width TFWHM vs. modulation depth (a) along the boundary GDD (b) in dependence on the saturation

Figure 7. Numerical spectral profiles for different modulation depths μ (inset shows the dependence of T FWHM). GDD

, δ = �0.05. The Lorentzian gain band of 5.3 THz (dashed line)

Theory of Laser Energy Harvesting at Femtosecond Scale http://dx.doi.org/10.5772/intechopen.75039 183

Chirped DS demonstrates a high potential for energy harvesting in both solid-state and fiber lasers [24, 25] due to enhanced stability provided by well-structured energy redistribution inside a pulse. An energy scalability results from the DS stretching that limits its peak power

not allow a substantial DS shortening.

corresponds to the stability threshold, <sup>ζ</sup> <sup>¼</sup> <sup>γ</sup>= 1.35 MW�<sup>1</sup>

corresponds to a Yb: YAG, the output energy Eout is of ≈ 0:011 μJ for 3% output coupler.

parameter ζ. Eout ≈ 0:011–0:014 μJ and other parameters correspond to Figure 7.

2.2. Chirped DS

Figure 7. Numerical spectral profiles for different modulation depths μ (inset shows the dependence of T FWHM). GDD corresponds to the stability threshold, <sup>ζ</sup> <sup>¼</sup> <sup>γ</sup>= 1.35 MW�<sup>1</sup> , δ = �0.05. The Lorentzian gain band of 5.3 THz (dashed line) corresponds to a Yb: YAG, the output energy Eout is of ≈ 0:011 μJ for 3% output coupler.

Figure 8. Pulse width TFWHM vs. modulation depth (a) along the boundary GDD (b) in dependence on the saturation parameter ζ. Eout ≈ 0:011–0:014 μJ and other parameters correspond to Figure 7.

Figure 9 illustrates the energy-dependence of the DS width and the stabilizing GDD for a low net-loss. The short pulses are possible in this case, as well, but a small modulation depth does not allow a substantial DS shortening.

#### 2.2. Chirped DS

devices. (i) a traditional ML mechanism uses the structured semiconductor devices (so-called semiconductor saturable absorber mirrors, SESAM) which (α) have a slow (~100 fs) response time; (β) a complicated and hardly-controllable kinematics including higher-order nonlinear effects, non-saturable losses, temperature and radiational damage, etc.; (γ) SAM and SPM effects are decoupled for this type of ML that requires growth of GDD for DS stabilization in accordance with the area theorem (Eq. (8)). (ii) As was shown (Eq. (11)), the minimum pulse width is

τΓ=μ p so that using the media with the broader gain band would provide a pulse shortening down to sub-100 fs [59]. However, such media with good optical quality are not widely

Nevertheless, an alternative approach to energy scalability of femtosecond pulses has been demonstrated in [7]. (i) ML mechanism can be provided by an instantaneous self-focusing (Kerr-lensing) induced by a set of nonlinear crystals inside a laser resonator. (ii) Such a mechanism combines both SAM and SPM that enhances the SAM parameters (μ and ζ) in parallel with the SPM one (γ, see Eq. (11)). As a result, the GDD value can be reduced in parallel with the DS shortening in agreement with the area theorem. (iii) A real-world gain

The Lorentzian gain profile can be taken into account by using the numerical simulations of

<sup>þ</sup> <sup>r</sup><sup>Ω</sup> 2 1 þ σ Ð j j a 2 dt

<sup>τ</sup> <sup>p</sup> .

As was shown in [60], the Lorentzian gain profile gives more efficient amplification and broader spectrum than the Gaussian one. Additionally, an inherent gain dispersion shifts the DS spectrum and affects its shape [62]. The numerically obtained pulse spectra for different modulation depths are shown in Figure 7. One can see a pronounced spectrum broadening

The dependences of minimum DS width and corresponding stabilizing GDD on the modulation depth for different values of the SAM saturation power ζ are shown in Figure 8. One can see that the DS shortens with μ in qualitative agreement with the analytical results presented above. Simultaneously, the DS spectrum is noticeably broader than the gain band that provides a generation of sub-50 fs pulses at the MW peak power level directly from an oscillator. Since a pulse is chirp-free, the threshold stabilizing anomalous GDD decreases, as well, in agreement with the soliton area theorem. Simultaneously, there is the nonmonotonic dependence of DS width on the SAM parameter ζ so that T decreases initially and then increases with the ζ-decrease, i.e., with the saturation power growth. The growth of saturation power (ζ-decrease) causes a threshold-like increase of pulse width and stabilizing GDD

� �

ðt

a z; t <sup>0</sup> � �

exp �Ω t � t <sup>0</sup> h i � �

dt0 , (17)

�∞

∝ ffiffiffiffiffiffiffiffiffiffiffi

182 High Power Laser Systems

<sup>∂</sup>za ¼ �Γ<sup>a</sup> <sup>þ</sup> μζj j <sup>a</sup> <sup>2</sup>

available and technologically advanced.

<sup>1</sup> <sup>þ</sup> <sup>ζ</sup>j j <sup>a</sup> <sup>2</sup> <sup>a</sup> � <sup>i</sup> <sup>β</sup><sup>2</sup>

for small modulation depths μ.

where a characteristic gain bandwidth is Ω ∝ 1= ffiffiffi

band profile is Lorentzian, not Gaussian as in Eqs. (9) and (14).

the generalized complex nonlinear Ginzburg-Landau equation [60, 61]:

a

and, correspondingly, pulse shortening with the modulation depth growth.

<sup>2</sup> <sup>∂</sup>t,ta <sup>þ</sup> <sup>γ</sup>j j <sup>a</sup> <sup>2</sup>

� �

Chirped DS demonstrates a high potential for energy harvesting in both solid-state and fiber lasers [24, 25] due to enhanced stability provided by well-structured energy redistribution inside a pulse. An energy scalability results from the DS stretching that limits its peak power and, thereby, suppresses an instability caused by nonlinearity. This factor is especially important for all-fiber lasers, where the strong contribution of nonlinear effects is inevitable with Tres-growth. VA predicts the following energy-scaling laws4 :

$$E \propto \left| \beta\_2 \right| / \mathbb{C} \sqrt{\pi \Gamma}, \quad E \propto \left| \beta\_2 \right| / \sqrt{\pi \Gamma} \tag{18}$$

For the SAM presented by Eq. (9), the adiabatic theory predicts the energy-scaling law in the

Since the chirped DS energy scaling is provided by its stretching ∝ψ, this process is reversible so that an output DS can be compressed by a factor ≈ 1=ψ. Nevertheless, some energy loss occurs with such compression due to nonuniformity of DS chirp [68]<sup>6</sup> that requires optimizing

The DS energy harvesting in both chirped and chirp-free regimes has a common problem of ML self-starting. The DS stability is a necessary but not sufficient condition of its existence because it must develop from some stochastic process in a laser (eventually, from a quantum noise). Existing theories of the ML self-start [32–42] predict that a lot of effects are involved in a pulse formation. However, a spontaneous formation of the DS from noise (the DS self-start) as a general problem has not been studied in depth. In optics, this is often considered as a technical issue, because here one can use one of the proven ML techniques to guarantee self-starting. After the initial kick, however, the DS evolves by itself, and recent experiments have shown controversial results: in high-power solid-state lasers, the strong oscillations (Q-switching) during the DS buildup dynamics hinders the DS self-start [69], while in a fiber laser, such oscillations can accelerate the self-start [70]. That is obviously connected to co-existence of nonlinearities with different time scales: instantaneous non-dissipative SPM, and non-instantaneous dissipative nonlinearities like stimulated Raman scattering (SRS), saturable absorber losses, and gain saturation. This issue is especially intriguing, as the dynamic gain saturation can provide a supple-

The growing nonlinearity results in quite nontrivial modification of dynamics [25] and causes whether DS stabilization or its chaotization [72–74]. For example, the practically relevant Yb-based thin-disk lasers possess reduced instantaneous nonlinearity and longer gain relaxation times as compared to a bulk Ti: sapphire laser. In the latter case, the enhanced dynamic gain saturation can destabilize a much-desired high-energy DS [75–77]. As another example, the experiments demonstrated, that DS energy scaling in all-fiber fiber lasers is limited by energy loss due to SRS [78]. Nevertheless, SRS could play a positive role providing the generation of dissipative Raman soliton and suppressing the optical turbulence [79–81]. The connection of this

phenomenon to the general issues of the turbulence theory waits for its exploration [82].

The adiabatic theory does not predict a spectral condensation near the carrier frequency ω ≈ 0 for this SAM law, but such a concentration is possible at spectrum edges. This phenomenon is clearly visible in the numerous experiments and can be

<sup>1</sup> � <sup>C</sup> <sup>p</sup> <sup>þ</sup> arctanh ffiffiffiffiffiffiffiffiffiffiffiffi

1 � C � � <sup>p</sup> � � (21)

Theory of Laser Energy Harvesting at Femtosecond Scale http://dx.doi.org/10.5772/intechopen.75039 185

ffiffiffiffiffiffiffiffiffiffiffiffi

form of [66]5

:

the DS and pulse compressor parameters.

mentary mechanism of DS formation [71].

explained by the DS perturbation theory [67].

That is a measure of the DS fidelity.

5

6

2.3. Main obstacles to the DS energy harvesting

<sup>E</sup><sup>∝</sup> <sup>β</sup><sup>2</sup> � � � � ζ ffiffiffi τ p 3 C

that gives the first expression in Eq. (18) in the C ! 0 (i.e., E ! ∞) limit.

for the SAM described by Eqs. (9) and (14), respectively [24, 25].

The chirped DS accumulates energy ∝ψ that allows using a so-called adiabatic theory for ψ ≫ 1 [24, 63, 64] which predicts a perfect energy scalability or a DS resonance [65] for the cubicquintic SAM (Eq. (14)). That means that energy can be scaled infinitely for C ¼ 1=3 due to pulse stretching and simultaneous spectral condensation:

$$\lim\_{\substack{\zeta \to 1/3\\ \Theta \to 1/3}} \begin{cases} E \to \infty \\ a^2 \to 1/\zeta \\ \Delta \to \sqrt{2\gamma/\beta\zeta} \\ \Theta \to 0 \text{ ("spectral condensation"} \end{cases} \tag{19}$$

where the DS spectral profile is described by a truncated Lorentzian function:

$$p(\omega) = \frac{6\pi\eta}{\kappa\zeta} \frac{H\{\Delta^2 - \omega^2\}}{\omega^2 + \Theta^2} \tag{20}$$

(here H is the Heaviside's function).

Figure 9. A low-loss regime with δ ¼ �0:05=3, ζ ¼ 100γ, 1% output coupler, μ = 2%.

<sup>4</sup> The negative sign of β<sup>2</sup> corresponds to a normal GDD in these notations.

For the SAM presented by Eq. (9), the adiabatic theory predicts the energy-scaling law in the form of [66]5 :

$$E \propto \frac{|\beta\_2|}{\zeta \sqrt{\pi}} \left[ \frac{3}{C} \sqrt{1 - C} + \text{arctanh} \left( \sqrt{1 - C} \right) \right] \tag{21}$$

that gives the first expression in Eq. (18) in the C ! 0 (i.e., E ! ∞) limit.

Since the chirped DS energy scaling is provided by its stretching ∝ψ, this process is reversible so that an output DS can be compressed by a factor ≈ 1=ψ. Nevertheless, some energy loss occurs with such compression due to nonuniformity of DS chirp [68]<sup>6</sup> that requires optimizing the DS and pulse compressor parameters.
