3. Femtosecond pulse enhancement in an external resonator

Using a high-finesse ER coupled with a mode-locked femtosecond laser is the method of energy storing allowing broadband absorption spectroscopy [83], high-harmonic generation and frequency comb generation up to the extreme ultraviolet frequency [84, 85].

Equations describing a coupling with ER are [86, 87]:

$$b\_r = r b\_{\rm in} + \Theta a\_{\rm in} \quad a\_r = \Theta b\_{\rm in} - r a\_{\rm in} \tag{22}$$

where br and bin are the reflected and incident fields on the coupler from a side of femtosecond oscillator; ar and ain are the corresponding fields from a side of ER; and <sup>θ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>r</sup><sup>2</sup> <sup>p</sup> , <sup>r</sup> are the transmittance and reflection coefficients of a coupler, respectively. The field inside ER evolves as:

$$a\_{\dot{m}}(t) = \exp\left[-\Gamma + i\phi - \beta\_1 \frac{\partial}{\partial t} + i \sum\_{m=2}^{N} \ddot{r}^m \beta\_m \frac{\partial^m}{\partial t^m} + i\gamma |a\_r(t)|^2\right] a\_r(t),\tag{23}$$

where Γ is a net-loss coefficient; ϕ, β<sup>1</sup> are the phase and group-velocity delays, respectively; β<sup>m</sup> are the mth-order GDD coefficients, and γ is the SPM coefficient.

In the absence of group-delay, GDD and SPM in ER, the energy, and power enhancement factors (Qe and Qp, respectively) are [87]:

$$Q\_{\epsilon} = Q\_{p} = \left(\frac{\theta}{1 - r \exp\left[-\Gamma\right]}\right)^{2}.\tag{24}$$

Eq. (26) demonstrates reducing the enhancement factors due to GDD [87]. Indeed, β<sup>m</sup> 6¼ 0

Figure 10. (a) Maximum Qp inside ER in the dependence of the third-order dispersion (TOD, m = 3) and the fourth-order dispersion (FOD, m = 4) for γ = 0, Δϕ = 0, β<sup>1</sup> ~ �0.5 fs, Γ = 0.5% and 25 fs sech-shaped incident pulse at 790 nm central

overlap between the pulses from a laser and ER. In combination with a chirp appearance and a

It is clear that destructive action of higher-order GDD (m > 2 in Eq. (25)) on enhancement factor of ER grows with the pulse shortening so that a thorough dispersion-engineering of ER mirrors

Additionally, the enhancement factor control can be provided by realizing a soliton-like regime in the nonlinear ER with γ 6¼ 0 (Figure 11) [87, 88]. In the absence of higher-order dispersions

γPð Þ0

<sup>2</sup> ,Δ<sup>ϕ</sup> ¼ � <sup>θ</sup>

m

P tð Þ <sup>p</sup> exp ð Þ iqz :

γPð Þ0

<sup>2</sup> , q <sup>¼</sup> <sup>0</sup>: (27)

Π � �<sup>2</sup>

� � that worsens the spectral

Theory of Laser Energy Harvesting at Femtosecond Scale http://dx.doi.org/10.5772/intechopen.75039 187

reduces the spectral power at the spectrum edges <sup>∝</sup> <sup>1</sup><sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup><sup>m</sup>β<sup>2</sup>

wavelength. (b) The corresponding optimal second-order dispersion (GDD, m = 2) [88].

within a sufficiently broad spectral range is required [28, 87, 88].

(i.e., <sup>m</sup> = 2), Eq. (25) has an exact soliton-like solution a tðÞ¼ ffiffiffiffiffiffiffiffi

, <sup>β</sup><sup>2</sup> ¼ � <sup>θ</sup><sup>T</sup>

Π � �<sup>2</sup>

This soliton-like pulse can be perturbed by higher-order dispersions (m > 2) induced by the broad-band ER mirrors so that optimization of ER parameters is required in this case, as well

A promising possibility of the Qp-increase results from a loss compensation by a gain inside ER. In this case, Eq. (25) has to be supplemented by the term <sup>r</sup>rτ∂t,ta with the modified <sup>Π</sup> <sup>¼</sup> <sup>1</sup> � <sup>r</sup> <sup>þ</sup> <sup>r</sup>ð Þ <sup>Γ</sup> � <sup>r</sup> <sup>9</sup> (see Eqs. (9) and (25)). One has to note, that ER remains below lasing threshold and a resonator mode in an active crystal (Ti: sapphire in our case) has to be sufficiently broad to suppress gain

sech <sup>t</sup> T � �<sup>2</sup>

P tðÞ¼ <sup>θ</sup>

8

8

9

(Figure 11).

Π � �<sup>2</sup>

Here β<sup>2</sup> < 0 corresponds to an anomalous dispersion.

r < Γ is the stability condition.

pulse broadening in ER, these factors drop both Qe and Qp (Figure 10).

Under the condition of weak changes of the field during single cavity round-trip, Eqs. (22) and (23) can be rewritten in the form of the generalized driven nonlinear Schrödinger equation for the intracavity field a(t) [88]:

$$\frac{\partial a(z,t)}{\partial z} = \left[ -\Pi + i\Delta\phi - \beta\_1 \frac{\partial}{\partial t} + i \sum\_{m=2} i^m \beta\_m \frac{\partial^m}{\partial t^m} + i\gamma |a|^2 \right] a + \mathcal{O}\Phi(t), \tag{25}$$

where z is a cavity round-trip number, Π ¼ 1 � r þ rΓ, Φ(t) is an incident field amplitude and Δϕ is a phase offset from the resonance ϕ = π. In the absence of SPM (vacuum ER) but with GDD induced by resonator mirrors, the solutions for the energy and power enhancement factors are7 :

$$Q\_{\epsilon} = \frac{1}{2\pi \int\_{-\infty}^{\infty} |\Phi(t)|^2 dt} \int\_{-\infty}^{\infty} \left| \frac{\int\_{-\infty}^{\infty} \Phi(t) e^{i\omega t}}{\Pi + i \left(\beta\_1 \omega - \beta\_2 \omega^2 + \beta\_3 \omega^3 - \beta\_4 \omega^4\right)} \right|^2 d\omega,$$

$$Q\_p = \max\left\{ \left| \frac{1}{2\pi} \int\_{-\infty}^{\infty} \left[ \frac{\int\_{-\infty}^{\infty} \Phi(t) e^{i\omega t} dt}{\Pi + i \left(\beta\_1 \omega - \beta\_2 \omega^2 + \beta\_3 \omega^3 - \beta\_4 \omega^4\right)} \right] e^{-i\omega t} d\omega \right|^2 \right\}. \tag{26}$$

<sup>7</sup> Γ ≪ 1 andm ≤ 4 are assumed. The field amplitude and the pulse width are normalized to those of incident.

Figure 10. (a) Maximum Qp inside ER in the dependence of the third-order dispersion (TOD, m = 3) and the fourth-order dispersion (FOD, m = 4) for γ = 0, Δϕ = 0, β<sup>1</sup> ~ �0.5 fs, Γ = 0.5% and 25 fs sech-shaped incident pulse at 790 nm central wavelength. (b) The corresponding optimal second-order dispersion (GDD, m = 2) [88].

Eq. (26) demonstrates reducing the enhancement factors due to GDD [87]. Indeed, β<sup>m</sup> 6¼ 0 reduces the spectral power at the spectrum edges <sup>∝</sup> <sup>1</sup><sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup><sup>m</sup>β<sup>2</sup> m � � that worsens the spectral overlap between the pulses from a laser and ER. In combination with a chirp appearance and a pulse broadening in ER, these factors drop both Qe and Qp (Figure 10).

It is clear that destructive action of higher-order GDD (m > 2 in Eq. (25)) on enhancement factor of ER grows with the pulse shortening so that a thorough dispersion-engineering of ER mirrors within a sufficiently broad spectral range is required [28, 87, 88].

Additionally, the enhancement factor control can be provided by realizing a soliton-like regime in the nonlinear ER with γ 6¼ 0 (Figure 11) [87, 88]. In the absence of higher-order dispersions (i.e., <sup>m</sup> = 2), Eq. (25) has an exact soliton-like solution a tðÞ¼ ffiffiffiffiffiffiffiffi P tð Þ <sup>p</sup> exp ð Þ iqz :

$$P(t) = \left(\frac{\theta}{\Pi}\right)^2 \text{sech}\left(\frac{t}{T}\right)^2, \beta\_2 = -\left(\frac{\theta \, T}{\Pi}\right)^2 \frac{\gamma P(0)}{2}, \Delta\phi = -\left(\frac{\theta}{\Pi}\right)^2 \frac{\gamma P(0)}{2}, q = 0. \tag{27}$$

8 This soliton-like pulse can be perturbed by higher-order dispersions (m > 2) induced by the broad-band ER mirrors so that optimization of ER parameters is required in this case, as well (Figure 11).

A promising possibility of the Qp-increase results from a loss compensation by a gain inside ER. In this case, Eq. (25) has to be supplemented by the term <sup>r</sup>rτ∂t,ta with the modified <sup>Π</sup> <sup>¼</sup> <sup>1</sup> � <sup>r</sup> <sup>þ</sup> <sup>r</sup>ð Þ <sup>Γ</sup> � <sup>r</sup> <sup>9</sup> (see Eqs. (9) and (25)). One has to note, that ER remains below lasing threshold and a resonator mode in an active crystal (Ti: sapphire in our case) has to be sufficiently broad to suppress gain

3. Femtosecond pulse enhancement in an external resonator

and frequency comb generation up to the extreme ultraviolet frequency [84, 85].

Equations describing a coupling with ER are [86, 87]:

186 High Power Laser Systems

ainðÞ¼ t exp �Γ þ iϕ � β<sup>1</sup>

<sup>∂</sup><sup>z</sup> ¼ �<sup>Π</sup> <sup>þ</sup> <sup>i</sup>Δ<sup>ϕ</sup> � <sup>β</sup><sup>1</sup>

�<sup>∞</sup> j j <sup>Φ</sup>ð Þ<sup>t</sup> <sup>2</sup>

2π ð ∞

� � � � � �

< :

�∞

dt ð ∞

�∞

� � � � �

factors (Qe and Qp, respectively) are [87]:

the intracavity field a(t) [88]:

∂a zð Þ ; t

Qe <sup>¼</sup> <sup>1</sup> 2π Ð <sup>∞</sup>

Qp <sup>¼</sup> max <sup>1</sup>

7

are the mth-order GDD coefficients, and γ is the SPM coefficient.

Using a high-finesse ER coupled with a mode-locked femtosecond laser is the method of energy storing allowing broadband absorption spectroscopy [83], high-harmonic generation

where br and bin are the reflected and incident fields on the coupler from a side of femtosecond

transmittance and reflection coefficients of a coupler, respectively. The field inside ER evolves as:

where Γ is a net-loss coefficient; ϕ, β<sup>1</sup> are the phase and group-velocity delays, respectively; β<sup>m</sup>

In the absence of group-delay, GDD and SPM in ER, the energy, and power enhancement

Under the condition of weak changes of the field during single cavity round-trip, Eqs. (22) and (23) can be rewritten in the form of the generalized driven nonlinear Schrödinger equation for

" #

where z is a cavity round-trip number, Π ¼ 1 � r þ rΓ, Φ(t) is an incident field amplitude and Δϕ is a phase offset from the resonance ϕ = π. In the absence of SPM (vacuum ER) but with GDD induced by resonator mirrors, the solutions for the energy and power enhancement factors are7

Ð ∞

Γ ≪ 1 andm ≤ 4 are assumed. The field amplitude and the pulse width are normalized to those of incident.

Ð ∞

�<sup>∞</sup> <sup>Φ</sup>ð Þ<sup>t</sup> <sup>e</sup><sup>i</sup>ω<sup>t</sup>

8 <sup>2</sup>

<sup>Π</sup> <sup>þ</sup> <sup>i</sup> <sup>β</sup>1<sup>ω</sup> � <sup>β</sup>2ω<sup>2</sup> <sup>þ</sup> <sup>β</sup>3ω<sup>3</sup> � <sup>β</sup>4ω<sup>4</sup> � � " #

�<sup>∞</sup> <sup>Φ</sup>ð Þ<sup>t</sup> <sup>e</sup><sup>i</sup>ω<sup>t</sup> <sup>Π</sup> <sup>þ</sup> <sup>i</sup> <sup>β</sup>1<sup>ω</sup> � <sup>β</sup>2ω<sup>2</sup> <sup>þ</sup> <sup>β</sup>3ω<sup>3</sup> � <sup>β</sup>4ω<sup>4</sup> � �

dt

Qe <sup>¼</sup> Qp <sup>¼</sup> <sup>θ</sup>

∂ ∂t þ i X m¼2 i <sup>m</sup>β<sup>m</sup> ∂m ∂t

m¼2 i <sup>m</sup>β<sup>m</sup> ∂m ∂t

1 � rexp ½ � �Γ � �<sup>2</sup>

" #

oscillator; ar and ain are the corresponding fields from a side of ER; and <sup>θ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

∂ ∂t þ i X N

br ¼ rbin þ θain, ar ¼ θbin � rain, (22)

<sup>m</sup> <sup>þ</sup> <sup>i</sup>γj j arð Þ<sup>t</sup> <sup>2</sup>

<sup>m</sup> <sup>þ</sup> <sup>i</sup>γj j <sup>a</sup> <sup>2</sup>

<sup>1</sup> � <sup>r</sup><sup>2</sup> <sup>p</sup> , <sup>r</sup> are the

arð Þt , (23)

: (24)

a þ θΦð Þt , (25)

� � � � �

e �iωt dω

2 dω,

> � � � � � �

9 = ;: :

(26)

<sup>8</sup> Here β<sup>2</sup> < 0 corresponds to an anomalous dispersion.

<sup>9</sup> r < Γ is the stability condition.

saturation and SPM. The soliton-like regime increases the enhancement factor and reduces the

Theory of Laser Energy Harvesting at Femtosecond Scale http://dx.doi.org/10.5772/intechopen.75039 189

In this work, the approaches to an energy harvesting at femtosecond scale are reviewed and elaborated theoretically with a close connection with both solid-state and fiber ML oscillators including, in particular, a nonlinear ER. The basic concept here is a dissipative soliton allowing an extra-energy and spectral width scaling under fine control of the laser parameters. This concept is a very productive for different applications and brings a high-energy physics in "physics laboratory" where extremal parameters result from not an onslaught but rather "subtle tuning." This tuning requires multi-disciplinary approaches providing the multi-scale power and energy harvesting, which application areas range from fundamental quantum mechanics to neuroscience and sociology, and include, in particular, a "quantum engineering" of Bose-Einstein and quasi-particle condensates. A further outlook concerns a study of nonlinear dynamics of complicated nonlinear systems far from thermodynamic equilibrium, which is based on their "metaphoric" modeling in more simple and controllable laser systems.

The author acknowledges the support from Austrian Science Fund (FWF project P24916).

[1] Südmeyer T, Marchese SV, Hashimoto S, Baer CRE, Gingras G, Witzel B, Keller U. Femtosecond laser oscillators for high-field science. Nature Photonics. 2008;2:599-604.

[2] Brabec T, editor. Strong Field Laser Physics. New York: Springer; 2008. 585 p. DOI: 10.1007/

Computational results have been achieved using the Vienna Scientific Cluster (VSC).

Address all correspondence to: vladimir.kalashnikov@tuwien.ac.at

Institute of Photonics, Vienna University of Technology, Vienna, Austria

sensitivity to higher-order dispersions in this case, as well (Figure 12).

4. Conclusions

Acknowledgements

Author details

References

Vladimir L. Kalashnikov

DOI: 10.1038/nphoton.2008.194

978-0-387-34755-4

Figure 11. Dependence of the maximum Qp on TOD and FOD (a) optimized by control of β<sup>1</sup> (c), β<sup>2</sup> (b), and the dimensionless SPM coefficient γPmax (d). Incident pulse TFWHM is 25 fs, Γ = 0.25%.

Figure 12. Qe (a) and the pulse width (b) in the presence of gain, SPM, and GDD in ER. Γ = 0.04, the width of the incident 100 nJ pulse is of 30 fs, the laser beam size in the 2 mm Ti: sapphire active crystal is 1.1 mm.

saturation and SPM. The soliton-like regime increases the enhancement factor and reduces the sensitivity to higher-order dispersions in this case, as well (Figure 12).
