3. Broadband optical parametric amplification

To get the population inversion between laser energy levels for laser amplification, we essentially need an efficient absorption of pump photons in Ti:sapphire crystals. The energy accumulated in the upper laser level can be the result of pumping with single or multiple pump laser beams. Angles between pump beams and seed pulse beam are noncritically defined and

Because the Ti atoms lifetime is in the range of few-μs, an acceptable delay between pump laser pulses and input stretched laser pulses is in the nanoseconds range. This temporal

Pulse duration of the recompressed pulse is inversely proportional to the optical frequency bandwidth which contains all phase-locked spectral components [20]. The highest amplification gain is obtained near the central wavelengths (790–800 nm) of the Ti:sapphire fluorescence spectrum, engendering the "gain narrowing" effect of the amplified laser pulse spectral band (Figure 2a). In the regenerative amplifiers and multi-pass amplifiers, with many passes through the laser amplifying media and high amplification factor, the effect of gain narrowing significantly contributes to the decrease of the spectral bandwidth of the amplified pulses

High-energy extraction efficiency can be obtained if laser amplifiers are working near the saturation regime, where the input laser pulse fluence is higher than the saturation fluence of the amplifying laser medium [20]. In this case, almost all accumulated energy on the upper level of the laser medium could be extracted and added to the input pulse energy [20]. The "red" spectral components travel in the leading edge of the temporally stretched pulse, whereas the "blue" spectral components are delayed in the trailing edge. In the amplifiers working near the saturation regime, due to the significant depletion of the upper laser-level population, the amplification factor of the "red" spectral components coming first in the amplifying medium is higher than that of the "blue" spectral components arriving on the trailing edge of the stretched pulse. The result is a redshift of the amplified laser pulse

Stretched amplified pulses are recompressed in a temporal stretcher with diffraction gratings, where "red" spectral components are delayed compared to the "blue" components. Both "gain narrowing" and "redshifting" effects contribute to the increase of the amplified pulse duration

The amplified spontaneous emission (ASE), which takes place in the laser media as long as the population inversion between the upper and lower laser levels exists, deteriorates the picosecond intensity contrast of femtosecond laser systems. By all Ti:sapphire amplification, it is very difficult to attain more than 1011 intensity contrast of femtosecond pulses, as it is required in

Dissipated heat in the active medium is given by the energy difference between the absorbed pump energy and the laser emitted energy. The thermal loading of the Ti:sapphire crystals produces beam wavefront distortions and phase dispersions of the spectral components of the large bandwidth laser pulses. It results in a poor beam focusing and an increase of the

are practically imposed by the amplifier geometry.

(Figure 2b).

46 High Power Laser Systems

after temporal recompression.

recompressed pulse duration.

case of PW-class femtosecond laser systems.

synchronization can be easily obtained with electronic devices.

spectrum, associated with a spectrum narrowing (Figure 2b and c).

Optical parametric amplification (OPA) is practically an instantaneous process without laser energy accumulation in the amplifying medium.

By absorption of pump photons with ω<sup>p</sup> frequency, the crystal molecules leave from their ground energy level E1 to an excited intermediate higher energy level E2 (Figure 3a). While an excited molecule returns to its initial ground state, a photon with ω<sup>s</sup> signal frequency and one "idler" photon with ω<sup>i</sup> = ω<sup>p</sup> - ω<sup>s</sup> are simultaneously created. This optical nonlinear process is very rapid compared to the signal and pump pulse duration.

A fraction of the pump beam energy is transferred to the signal beam. At the output of the nonlinear crystal, we get an amplified signal beam, a new generated idler beam, and a residual pump beam (Figure 3b).

Amplification takes place only if the seed pulse and the pump pulse are spatially and temporally overlapped in the nonlinear crystal, in a collinear (Figure 3c) or a noncollinear geometry (Figure 3d). In case of nanosecond pulses OPA, temporal overlapping can be obtained by electronic synchronization of the pump pulsed laser with the signal pulses. In case of femtosecond/picosecond pulses, temporal overlapping can be obtained only by optical synchronization of the interacting laser pulses.

The parametric amplification is produced under conditions of photon energy conservation and wave-vector phase matching, only for a certain orientation of the crystal and for well-defined angles between the wave vectors of the interacting laser beams (Figure 3c, d)

$$\begin{array}{l}\boldsymbol{\omega}\_{p} = \boldsymbol{\omega}\_{s} + \boldsymbol{\omega}\_{i} \\ \stackrel{\rightarrow}{k}\_{p} = \stackrel{\rightarrow}{k}\_{s} + \stackrel{\rightarrow}{k}\_{i} \end{array} \tag{1}$$

process: signal, pump and idler wavelengths, the angle between pump wave-vector and the crystal optical axis (θ), the angle between signal and pump wave-vectors (α), and the angle between signal and idler wave-vectors (β). For a monochromatic noncollinear parametric interaction, three parameters are free-chosen, usually signal wavelength, λs, pump wavelength, λp, and α angle between signal and pump beams. The idler wavelength (λi), θ, and β angles of a NOPA process in a certain nonlinear crystal can be calculated using the phase-

> sin <sup>α</sup> � nið Þ <sup>λ</sup><sup>i</sup> λi

cos <sup>α</sup> � nsð Þ <sup>λ</sup><sup>s</sup> λs

Under approximations of small initial signal beam intensity, without input idler beam, and

where L is the length of the nonlinear crystal, Is(0) is the input signal beam intensity, Is(L) is the

effective nonlinear coefficient, np,s,i are refractive indexes, ε<sup>0</sup> is the permittivity of free space, c is the speed of light, and Δk = kp � ks � ki is the wave-vector mismatch. The full width at half maximum (FWHM) phase-matching bandwidth is usually defined as the spectral range where the parametric gain Gs(Δk) is at least 50% from the peak gain obtained in the case of exact

2

Broad gain bandwidth can be obtained if, near the exact phase-matching condition, the wave-vector mismatch slowly varies depending on the signal wavelength. The Δk phase mismatch can be represented by Taylor series around the phase-matching signal frequency

> 1 3!

∂<sup>3</sup>Δk ∂ω<sup>3</sup> s � �

2!

… <sup>¼</sup> <sup>Δ</sup>kð Þ<sup>0</sup> <sup>þ</sup> <sup>Δ</sup><sup>k</sup>

ωs<sup>0</sup> ð Þ <sup>d</sup>ω<sup>s</sup> <sup>3</sup> þ

∂<sup>2</sup>ks ∂ω<sup>2</sup> s þ ∂<sup>2</sup>ki ∂ω<sup>2</sup> i

!

ð Þ<sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>k</sup>

, <sup>Γ</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup> <sup>ω</sup>sωid<sup>2</sup>

Gsð Þ¼ <sup>Δ</sup><sup>k</sup> <sup>1</sup>

Isð Þ<sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>2</sup> sinh<sup>2</sup>

eff Ip

sin β ¼ 0

� nið Þ <sup>λ</sup><sup>i</sup> λi

cos β ¼ 0

High-Power, High-Intensity Contrast Hybrid Femtosecond Laser Systems

http://dx.doi.org/10.5772/intechopen.70708

ð Þ gL

<sup>g</sup><sup>2</sup> (3)

nsninpε0c<sup>3</sup> , Ip is the pump beam intensity, deff is the

Gsð Þ Δk ¼ 0 (4)

ð Þ Δω <sup>2</sup> �

ð Þ<sup>2</sup> <sup>þ</sup> <sup>Δ</sup><sup>k</sup>

ð Þ<sup>3</sup> <sup>þ</sup> <sup>Δ</sup>kð Þ<sup>4</sup> <sup>þ</sup> …

(5)

(2)

49

matching Eqs. [21]

1 λp ¼ 1 λs þ 1 λi

output signal intensity, <sup>g</sup><sup>2</sup> <sup>¼</sup> <sup>Γ</sup><sup>2</sup> � <sup>Δ</sup><sup>k</sup>

phase-matching, Gs(Δk = 0) [21]

ωs<sup>0</sup> [22]

þ 1 4!

� 1 3!

<sup>Δ</sup><sup>k</sup> <sup>¼</sup> <sup>Δ</sup>kð Þ<sup>0</sup> <sup>þ</sup> <sup>∂</sup>Δ<sup>k</sup>

∂<sup>4</sup>Δk ∂ω<sup>4</sup> s � �

> ∂<sup>3</sup>ks ∂ω<sup>3</sup> s � <sup>∂</sup><sup>3</sup>ki ∂ω<sup>3</sup> i

∂ω<sup>s</sup> � � ωs<sup>0</sup> dω<sup>s</sup> þ 1 2!

ωs<sup>0</sup>

!

np λp; θ � � λp

np λp; θ � � λp

neglected pump beam depletion, the parametric gain is given by [21]

2 � �<sup>2</sup>

∂<sup>2</sup>Δk ∂ω<sup>2</sup> s � �

> ∂<sup>4</sup>ks ∂ω<sup>4</sup> s þ ∂<sup>4</sup>ki ∂ω<sup>4</sup> i

ð Þ <sup>d</sup>ω<sup>s</sup> <sup>4</sup> <sup>þ</sup> … <sup>≈</sup> <sup>Δ</sup>kð Þ<sup>0</sup> � <sup>∂</sup>ks

ð Þ Δω <sup>3</sup> � <sup>1</sup> 4! ωs<sup>0</sup>

∂ω<sup>s</sup>

!

ð Þ <sup>d</sup>ω<sup>s</sup> <sup>2</sup> <sup>þ</sup>

� <sup>∂</sup>ki ∂ω<sup>i</sup> � �Δω � <sup>1</sup>

ð Þ Δω <sup>4</sup>

Gsð Þ¼ <sup>L</sup> Isð Þ� <sup>L</sup> Isð Þ<sup>0</sup>

where kj ! , j = p,s,i, are the wave vectors of the pump, signal, and idler beams.

The host crystal of the parametric process is transparent to the interacting waves, and the amplification takes place without thermal loading of the nonlinear crystal.

Exact phase matching condition can be fulfilled only by monochromatic waves. Three beam parameters and three geometrical parameters are involved in a noncollinear OPA (NOPA)

Figure 3. Optical parametric amplification in nonlinear crystals. (a) OPA energy level diagram. (b) Principle of OPA in a nonlinear crystal. (c) Collinear OPA geometry. (d) Noncollinear OPA (NOPA) geometry.

process: signal, pump and idler wavelengths, the angle between pump wave-vector and the crystal optical axis (θ), the angle between signal and pump wave-vectors (α), and the angle between signal and idler wave-vectors (β). For a monochromatic noncollinear parametric interaction, three parameters are free-chosen, usually signal wavelength, λs, pump wavelength, λp, and α angle between signal and pump beams. The idler wavelength (λi), θ, and β angles of a NOPA process in a certain nonlinear crystal can be calculated using the phasematching Eqs. [21]

one "idler" photon with ω<sup>i</sup> = ω<sup>p</sup> - ω<sup>s</sup> are simultaneously created. This optical nonlinear

A fraction of the pump beam energy is transferred to the signal beam. At the output of the nonlinear crystal, we get an amplified signal beam, a new generated idler beam, and a residual

Amplification takes place only if the seed pulse and the pump pulse are spatially and temporally overlapped in the nonlinear crystal, in a collinear (Figure 3c) or a noncollinear geometry (Figure 3d). In case of nanosecond pulses OPA, temporal overlapping can be obtained by electronic synchronization of the pump pulsed laser with the signal pulses. In case of femtosecond/picosecond pulses, temporal overlapping can be obtained only by optical synchroniza-

The parametric amplification is produced under conditions of photon energy conservation and wave-vector phase matching, only for a certain orientation of the crystal and for well-defined

ω<sup>p</sup> ¼ ω<sup>s</sup> þ ω<sup>i</sup>

The host crystal of the parametric process is transparent to the interacting waves, and the

Exact phase matching condition can be fulfilled only by monochromatic waves. Three beam parameters and three geometrical parameters are involved in a noncollinear OPA (NOPA)

Figure 3. Optical parametric amplification in nonlinear crystals. (a) OPA energy level diagram. (b) Principle of OPA in a

nonlinear crystal. (c) Collinear OPA geometry. (d) Noncollinear OPA (NOPA) geometry.

! (1)

kp ! ¼ks ! þ ki

, j = p,s,i, are the wave vectors of the pump, signal, and idler beams.

angles between the wave vectors of the interacting laser beams (Figure 3c, d)

amplification takes place without thermal loading of the nonlinear crystal.

process is very rapid compared to the signal and pump pulse duration.

pump beam (Figure 3b).

48 High Power Laser Systems

where kj !

tion of the interacting laser pulses.

$$\begin{aligned} \frac{1}{\lambda\_p} &= \frac{1}{\lambda\_s} + \frac{1}{\lambda\_i} \\ \frac{n\_p \left(\lambda\_p, \Theta\right)}{\lambda\_p} \sin \alpha - \frac{n\_i(\lambda\_i)}{\lambda\_i} \sin \beta &= 0 \\ \frac{n\_p \left(\lambda\_p, \Theta\right)}{\lambda\_p} \cos \alpha - \frac{n\_s(\lambda\_s)}{\lambda\_s} - \frac{n\_i(\lambda\_i)}{\lambda\_i} \cos \beta &= 0 \end{aligned} \tag{2}$$

Under approximations of small initial signal beam intensity, without input idler beam, and neglected pump beam depletion, the parametric gain is given by [21]

$$\mathcal{G}\_s(L) = \frac{I\_s(L) - I\_s(0)}{I\_s(0)} = I^2 \frac{\sinh^2(\mathcal{g}L)}{\mathcal{g}^2} \tag{3}$$

where L is the length of the nonlinear crystal, Is(0) is the input signal beam intensity, Is(L) is the output signal intensity, <sup>g</sup><sup>2</sup> <sup>¼</sup> <sup>Γ</sup><sup>2</sup> � <sup>Δ</sup><sup>k</sup> 2 � �<sup>2</sup> , <sup>Γ</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup> <sup>ω</sup>sωid<sup>2</sup> eff Ip nsninpε0c<sup>3</sup> , Ip is the pump beam intensity, deff is the effective nonlinear coefficient, np,s,i are refractive indexes, ε<sup>0</sup> is the permittivity of free space, c is the speed of light, and Δk = kp � ks � ki is the wave-vector mismatch. The full width at half maximum (FWHM) phase-matching bandwidth is usually defined as the spectral range where the parametric gain Gs(Δk) is at least 50% from the peak gain obtained in the case of exact phase-matching, Gs(Δk = 0) [21]

$$\mathbf{G}\_s(\Delta k) = \frac{1}{2}\mathbf{G}\_s(\Delta k = 0) \tag{4}$$

Broad gain bandwidth can be obtained if, near the exact phase-matching condition, the wave-vector mismatch slowly varies depending on the signal wavelength. The Δk phase mismatch can be represented by Taylor series around the phase-matching signal frequency ωs<sup>0</sup> [22]

$$\begin{split} \Delta k &= \Delta k^{(0)} + \left(\frac{\partial \omega k}{\partial \omega\_i}\right)\_{\
u \alpha} d\omega\_s + \frac{1}{2!} \left(\frac{\partial^2 \Delta k}{\partial \omega\_s^2}\right)\_{\
u \alpha} (d\omega\_s)^2 + \frac{1}{3!} \left(\frac{\partial^3 \Delta k}{\partial \omega\_s^3}\right)\_{\
u \alpha} (d\omega\_s)^3 + \\ &+ \frac{1}{4!} \left(\frac{\partial^4 \Delta k}{\partial \omega\_s^4}\right)\_{\
u \alpha} (d\omega\_s)^4 + \dots \simeq \Delta k^{(0)} - \left(\frac{\partial k\_s}{\partial \omega\_s} - \frac{\partial k\_i}{\partial \omega\_i}\right) \Delta \omega - \frac{1}{2!} \left(\frac{\partial^2 k\_s}{\partial \omega\_s^2} + \frac{\partial^2 k\_i}{\partial \omega\_i^2}\right) (\Delta \omega)^2 - \\ &- \frac{1}{3!} \left(\frac{\partial^3 k\_s}{\partial \omega\_s^3} - \frac{\partial^3 k\_i}{\partial \omega\_i^3}\right) (\Delta \omega)^3 - \frac{1}{4!} \left(\frac{\partial^4 k\_s}{\partial \omega\_s^4} + \frac{\partial^4 k\_i}{\partial \omega\_i^4}\right) (\Delta \omega)^4 \dots = \Delta k^{(0)} + \Delta k^{(1)} + \Delta k^{(2)} + \Delta k^{(3)} + \Delta k^{(4)} + \dots \end{split} \tag{5}$$

where Δk = 0 represents the condition for quasi-monochromatic phase matching; Δk (0) = Δk (1) = 0 is the condition for optical parametric broad gain bandwidth.

An ultra-broad bandwidth (UBB) of phase-matching can be obtained for Δk (0) = Δk (1) = Δk (2) = 0. In this case, two more equations must be added to the three-equation system (2) [22]

$$\begin{aligned} \nu\_{\mathcal{S}^s} &= \upsilon\_{\mathcal{S}^i} \cos \beta\\ \frac{\partial^2 k\_s}{\partial \omega\_s^2} \cos \beta + \frac{\partial^2 k\_i}{\partial \omega\_i^2} - \frac{\sin^2 \beta}{\upsilon\_{\mathcal{S}^s}^2 k\_i} &= 0 \end{aligned} \tag{6}$$

4. Optical parametric chirped pulse amplification

laser pulses.

parametric amplification process.

OPCPA was proposed as an alternative solution for the amplification of large bandwidth stretched laser pulses [24] (Figure 5). Drawbacks of the Ti:sapphire CPA, particularly those related to the amplified spectral band narrowing, intensity contrast decrease, and thermal loading, can be overcome in OPCPA laser systems. Signal pulses generated by a broad bandwidth femtosecond oscillator are temporally stretched and synchronized to the pump pulses. Signal and pump pulses have similar durations, usually in the range of picoseconds or nanoseconds. The pump laser wavelength is chosen among the available high-energy green nanosecond lasers, such as frequency-doubled Nd:YAG (532 nm), Nd:glass (527 nm), Yb:YAG (515 nm) lasers. After OPCPA in one or more amplifier stages with nonlinear crystals, enhanced signal pulses can be temporally recompressed to get higher power femtosecond

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51

Unlike CPA, OPCPA is free from gain narrowing and redshifting effects. Because the host crystal is transparent to the interacting beams, thermal loading is practically absent in the

On the other hand, in the case of OPCPA, the spectrum of the amplified laser pulse is sensitive to the angle between signal and pump laser beams. The parametric amplification of each signal spectral component depends on the local instantaneous pump radiation intensity. In order to keep a stable amplified signal spectrum from pulse to pulse, high temporal and spatial stability

Unlike CPA amplifiers, due to angular constraints between pump and signal wave vectors, imposed by the unique phase-matching geometry, in OPCPA experimental setups usually a single pump laser beam can be used (Figure 6). To amplify broadband chirped laser pulses, laser systems based on noncollinear OPCPA (NOPCPA) configuration, imposed by the condi-

within ~1 ns pulse duration, is required. It is a real challenge to build a single-beam laser able

–103 J pump energy,

tions of UBB parametric amplification in nonlinear crystals, were developed [3, 25–29].

of the pump beams, as well as very stable experimental setup, are required.

For high-energy final amplifiers of multi-PW laser systems, as much as 10<sup>2</sup>

to deliver the pump pulses for these high-energy OPCPA stages.

Figure 5. Principle of optical parametric chirped pulse amplification.

where vgs and vgi are group velocities of signal wave and idler wave, respectively.

Particularly in the case of high-energy laser pulse amplification, only a couple of existing highenergy lasers are suitable for OPA pumping. For this reason, usually the pump laser wavelength λ<sup>p</sup> represents the free-chosen parameter of the OPA process. For a certain nonlinear crystal, the other five parameters, including signal central wavelength, are deduced from the five-equation system comprising Eqs. (2) and (6).

UBBs of more than 100 nm, able to support amplification of sub-10-fs laser pulses, can be obtained in nonlinear crystals [22], like potassium dideuterium phosphate (DKDP) and BBO. Ultra-broad gain bandwidths for BBO and DKDP crystals, pumped by green nanosecond lasers, in NOPA configuration are shown in Figure 4. Gain bandwidths were calculated assuming plane interacting waves, uniform pump intensity distribution, no input idler beam, and negligible pump beam intensity depletion. For both NOPA processes, I considered a flat pump intensity IP of 1 GW/cm<sup>2</sup> , which can be accepted without damage risk of currently used nonlinear crystals in case of about one-nanosecond pump pulse duration (e.g., the data sheets of the manufacturing company Altechna) [23]. Different lengths were considered for DKDP and BBO crystals, corresponding to similar gain values in the parametric amplification process.

The UBB phase-matching of DKDP crystals is centered around λS0 = 900 nm central wavelength, whereas the UBB of BBO crystals is centered in the range of 800 nm wavelength, practically overlapped to the gain bandwidth of Ti:sapphire laser media.

Figure 4. NOPA gain spectra. IP = 1 GW/cm<sup>2</sup> . (a) 80-mm-long DKDP crystal; λP(DKDP) = 0.527 μm, θDKDP = 37.0�, αDKDP = 0.92�, λS0 = 900 nm; UBBDKDP ≈ 135 nm. (b) 10-mm-long BBO crystal, λP(BBO) = 0.532 μm, θBBO = 23.8�, αBBO = 2.4�, λS0 = 0.825 μm; UBBBBO ≈ 150 nm.
